;-NRLF 


St    Dlt, 


BRM 


REESE  LIBRARY 


-- n    n    i\ 


UNIVERSITY  OF  CALIFORNIA. 


L 


Accessions  No. 


•    I- 


?4fl>X 

HE  \ 

{ TTNIVERSITY) 
^ 


PRACTICAL  TREATISE 


ON 


GEARING 


FIFTH    EDITION. 


PROVIDENCE,  R.  I. 

BROWN  &  SHARPE  MANUFACTURING  COMPANY, 
1896, 


Entered  according  to  Act  of  Congress,  in  the  year  1896  by 

BROWN  &  SHARPE  MFG.  CO., 

In  the  Office  of  the  Librarian  of  Congress  at  Washington. 

Registered  at  Stationers'  Hall,  London,  Eng. 

All  rights  reserved. 


• 


This  Book  is  made  for  men  in  practical  life;  for  those  who 
would  like  to  know  how  to  construct  gear  wheels,  but  whose 
duties  do  not  afford  them  sufficient  leasure  for  acquiring  a 
technical  knowledge  of  the  subject. 


CONTENTS. 


PART     I. 
CHAPTER   I. 

PAGE 

Pitch   Circle — Pitch — Tooth — Space — Addendum  or  Face — 

Flank — Clearance 1 

CHAPTER  II. 
Classification — Sizing  Blanks  and  Tooth  Parts  from  Circular 

Pitch — Center  Distance 5 

CHAPTER  III. 
Single  Curve  Gears  of  30  Teeth  and  over 9 

CHAPTER  IV. 
Hack  to  Mesh  with  Single  Curve  Gears  having  30  Teeth  and 

over. 12 

CHAPTER  V. 
Diametral  Pitch — Sizing  Blanks  and  Teeth — Distance  between 

Centers  of  Wheels 1G 

CHAPTER  VI. 
Single-Curve  Gears,  having  Less  than  30  Teeth — Gears  and 

Backs  to  Mesh  with  Gears  having  Less  than  30  Teeth. . .     20 

CHAPTER  VII. 
Double-Curve  Teeth— Gear  of  15  Teeth— Rack 25 

CHAPTER  VIII. 
Double-Curve  Gears,  having  More  and  Less  than  15  Teeth — 

Annular  Gears 30 

• 

CHAPTER  IX. 
Bevel  Gear  Blanks.  , 34 

CHAPTER  X. 
Bevel  Gears— Form  and  Size  of  Teeth— Cutting  Teeth 41 


VI  CONTENTS. 

CHAPTER  XI. 

PAGE. 

Worm  Wheels — Sizing  Blanks  of  32  Teeth  and  over. . , 62 

CHAPTER  XII. 

Sizing  Gears  when  the  Distance  between  Centers  and  the 
Katio  of  Speeds  are  fixed — General  Kemarks — Width  of 
Face  of  Spur  Gears — Speed  of  Gear  Cutters — Table  of 
Tooth  Parts..  77 


PART     II. 

CHAPTER   I. 
Tangent  of  Arc  and  Angle 

CHAPTER  II. 
Sine,    Cosine   and   Secant — Some    of   their    Applications   in 

Macnine  Construction 97 

CHAPTER  III. 

Application  of  Circular  Functions— Whole  Diameter  of  Bevel  Gear 

Blanks— Angles  of  Bevel  Gear  Blanks 1°6 

CHAPTER  IV 
Spiral  Gears— Calculations  for  Pitch  of  Spirals 113 

CHAPTER  V. 

Examples  in  Calculations  of  Pitch  of  Spirals  —  Angle  of 
Spiral— Circumference  of  Spiral  Gears— A  few  Hints  on 
Cutting 117 

CHAPTER  VI. 

Normal  Pitch  of  Spiral  Gears— Curvature  of  Pitch  Surface- 
Formation  of  Cutters !  22 


CONTENTS.  VII 

CHAPTER  VII. 

PAGEo 

Screw  Gears  and  Spiral  Gears — General  Kemarks. . . , 128 

CHAPTER  VIII. 

Continued   Fractions — Some  Applications  in  Machine  Con- 
struction.      .    .    .    . 0 0.0.         130 

CHAPTER  IX. 
Angle  of  Pressure , .  137 

CHAPTER  X. 
Internal  Gears — Tables — Index t, „ . . .    139 

CHAPTER  XL 
Strength  of  Gears — Tables , , . . .     142 


"  •«S»|N^ 

^v 

XTNIVERSITY) 

OF 


/' 


^ 


PART     I. 


CHAPTER  I. 

PITCH  CIRCLE,  PITCH,  TOOTH,  SPACE,  ADDENDUM  OR  FACE,  FLANK, 

CLEARANCE. 


Let  two  cylinders,  Fig.  1,  touch  each  other,  their  Original  Cyi- 
axes  be  parallel  and  the  cylinders  be  on  shafts,  turning 
freely.  If,  now,  we  turn  one  cylinder,  the  adhesion  of 
its  surface  to  the  surface  of  the  other  cylinder  will 
make  that  turn  also.  The  surfaces  touching  each 
other,  without  slipping  one  upon  the  other,  will  evi- 
dently move  through  the  same  distance  in  a  given 

J          .  .  Linear  Veloci- 

time.     This  surface  speed  is  called  linear  velocity.         ty. 

TANGENT    CYLINDERS. 


LINEAE  VELOCITY  is  the  distance  a  point  moves  along 
a  line  in  a  unit  of  time. 

The  line  described  by  a  point  in  the  circumference 
of  either  one  of  these  cylinders,  as  it  rotates,  may  be  call- 
ed an  arc.  The  length  of  the  arc  (which  may  be  greater 
or  less  than  the  circumference  of  cylinder),  described 
in  a  unit  of  time,  is  the  velocity.  The  length,  expressed 
in  linear  units,  as  inches,  feet,  etc.,  is  the  linear  velocity. 


BROWN    &    SHARPS    MFG.    CO. 


iocity!ular  Ve 


The  length,  expressed  in  angular  units,  as  degrees,  is 
the  angular  velocity. 

If  now,  instead  of  1°  we  take  360°,  or  one  turn,  as 
^e  angular  unit?  and  1  minute  as  the  time  unit,  the 
angular  velocity  will  be  expressed  in  turns  or  revolu- 
tions per  minute. 

If  these  two  cylinders  are  of  the  same  size,  one  will 
make  the  same  number  of  turns  in  a  minute  that  the 

Relative  An-  other  makes.     If  one  cylinder  is  twice  as  lame  as  the 
gular  Velocity.  •* 

.     other,  the  smaller  will  make  two  tuins  while  the  larger 
makes  one,  but  the  linear  velocity  of  the  surface  of 

.     each  cylinder  remains  the  same. 

This  combination  would  be  very  useful  in  mechan- 
ism if  we  could  be  sure  that  one  cylinder  would  always 
turn  the  other  without  slipping. 


In  the  periphery  of  these  two  cylinders,  as  in  Fig. 
2,  cut  equidistant  grooves.  In  any  grooved  piece  the 
places  between  grooves  are  called  lands.  Upon  the 
lands  add  parts  ;  these  parts  are  called  addenda.  A 
land  and  its  addendum  is  called  a  tooth.  A  toothed 
cylinder  is  called  a  gear.  Two  or  more  gears  with 
teeth  interlocking  are  called  a  train.  A  line,  c  c',  Fig. 


PROVIDENCE,    R.    I. 


Addendum 

Circle. 


"2  or  3,  between  the  centers  of  two  wheels  is  called  the    Line  of  Cen_ 
line  of  centers.     A.  circle  just  touching  the  addenda tej 
is  called  the  addendum  circle. 

The  circumference  of  the  cylinders  without  teeth  is 
called   the  pitch  circle.     This  circle  exists  geometri- pitch  circle- 
cally  in  every  gear  and  is  still  called  the  pitch  circle    pitch    circle 
or  the  primitive  circle.    In  the  study  of  gear  wheels,  it  jjj 
is  the  problem  to  so  shape  the  teeth  that  the   pitch  circle 
circles  will  just  touch  each  other  without  slipping. 

On  two  fixed  centers  there  can  turn  only  two  circles, 
one  circle  on  each  center,  in  a  given  relative  angular 
velocity  and  touch  each  other  without  slipping. 


THICKNESS  OF     j 

TOOTH 
AT  PITCH  LINE 


Fig. 


GA 


BROWN    &    SHARPE   MFG.    CO. 


tiots  bofepart8 

for  Teeth  and 


Space.  The  groove  between  two  teeth  is  called  a  space- 

In  cut  gears  the  width  of  space  at  pitch  line  and  thick- 
ness of  tooth  at  pitch  line  are  equal.  The  distance 
between  the  center  of  one  tooth  and  the  center  of  the 

circular  Pitch,  next  tooth,  measured  along  the  pitch  line,  is  the  cir- 
cular pitch;  that  is,  the  circular  pitch  is  equal  to  a- 
Tooth  Thick-  tooth  and  a  space  ;  hence,  the  thickness  of  a  tooth  at 
the  pitch  line  is  equal  to  one-half  the  circular  pitch. 

-D—  diameter  of  addendum  circle. 

J)'  —          "  "  pitch  " 

P'=:  circular  pitch. 
"     £—  thickness  of  tooth  at  pitch  line. 
"     s=  addendum  or   face,  also  length  of  working^ 

part  of  tooth  below  pitch  line  or  flank. 
"    2s  —  D"   or  twice  the    addendum,    equals    the 

working  depth  of  teeth   of   two  gears   in 

mesh. 
"    f=.  clearance   or   extra   depth   of   space   below 

working  depth. 

"    s  +f=  depth  of  space  below  pitch  line. 
"    D"+/=  whole  depth  of  space. 
"    N=  number  of  teeth  in  one  gear. 
"    7T—  3.1416  or  the  circumference  when  diameter 

is  1. 

P'  is   read    "P  prime.''     D"  is   read   "D    second.'' 
TC  is  read,  "  pi." 

If  we  multiply  the  diameter  of  any  circle  by  TT,  the 
TO  find   the  product  will  be  the  circumference  of  this  circle.     If 
and  UDiameter  we  divide  the  circumference  of  any  circle  by  TT,  the 
Cle'     quotient  will  be  the  diameter  of  this  circle. 


PROVIDENCE,  R.  I. 


CHAPTER  II. 

CLASSIFICATION-SIZING  BLANKS  AND  TOOTH  PARTS  FROM 
CIRCULAR  PITCH— CENTRE  DISTANCE— PATTERN  GEARS. 


If  we  conceive  the  pitch  of  a  pair  of  gears  to  te'yJrSSf1  °f 
made  the  smallest  possible,  we  ultimately  come  to  the 
conception  of  teeth  that  are  merely  lines  upon  the 
original  pitch  surfaces.  These  lines  are  called  ele- 
ments of  the  teeth.  Gears  may  be  classified  with 
reference  to  the  elements  of  their  teeth,  and  also  with 
reference  to  the  relative  position  of  their  axes  or  -shafts. 
In  most  gears  the  elements  of  teeth  are  either  straight 
lines  or  helices  (screw-like  lines). 

In  PART  I.  of  this  work,  we  shall  treat  upon  THREE 

KINDS  OF  GEARS  : 

First  —  SPUR  GEARS  ;  those  connecting  parallel  shafts  SPUI>  Gears- 
.and  whose  tooth  elements  are  straight. 

Second  —  BEVEL  "GEARS;  those  connecting  shafts  Bevel  Gears- 
whose  axes  meet  when  sufficiently  prolonged,  and  the 
elements  of  whose  teeth  are  straight  lines.  In  bevel 
gears  the  surfaces  '  that  touch  each  other,  without 
slipping,  are  upon  cones  or  parts  of  cones  whose 
apexes  are  at  the  same  point  where  axes  of  shafts  meet. 

Third  —  SCREW   OR  WORM   GEARS;    those  connecting  w®££  ^anL* 
shafts  that  are  neither  parallel  nor  meet,  and  the  ele- 
ments of  whose  teeth  are  helical  or  screw-like. 

The  circular  pitch  and  number  of  teeth  in  a  wheel  sizin{? 
being  given,  the   diameter  of  the  wheel  and  size  of      Blanks,  &c. 
tooth  parts  are  found  as  follows  : 

Dividing  by  3.1416  is  the  same  as  multiplying  by 
3.ili6~-3183;  hence,  multiply  the  cir- 


cumference of  a  circle  by  .3183  and  the  product  will  be 
the  diameter  of  the  circle.  Multiply  the  circular  pitch 
by  .3183  and  the  product  will  be  the  same  part  of  the 


6  BKOWN    &    SHAKPE    MFG.    CO. 

diameter  of  pitch  circle  that  the  circular  pitch  is  of  the 
circumference  of  pitch  circle.  This  part  or  modulus 
A  Diameter  is  called  a  diameter  pitch.  There  are  as  many  diameter 
pitches  contained  in  the  diameter  of  pitch  circle  as- 
there  are  teeth  in  the  wheel. 

pitch  Dam?ethe     Most  mechanics  make  the  addendum  of  teeth  equal 
measVre'thB*0  on©  Diameter  pitch.      Hence  we  can  designate  this 
same,  radially,  modulus  or  diameter  pitch  by  the  same  letter  as  we  do 
the  addendum;  that  is,  let  s  =  a  diameter  pitch. 

.3183  P'=s,  or  circular  pitch  multiplied  by  .3183=3,, 
or  a  diameter  pitch. 

Ns=D',  or  number  of  teeth  in  a   wheel,  multiplied 
PiteS^SSe.  °f  ^7  a  diameter  pitch,  equals  diameter  of  pitch  circle. 

(N  +  2)  s=D,  or  add  2  to  the  number  of  teeth,  mul- 
Wnoie  Diam- tiply  the  sum  by  s  and  product  will   be   the  whole 

eter.  ,5  * 

diameter. 

A— y,  or  one  tenth  of  thickness  of  tooth  at  pitch-line 
Clearance.       equals  amount  added  to  bottom  of  space  for  clearance. 
Some  mechanics  prefer  to  make /equal  to  ^  of  the 
working  depth  of  teeth,   or  .0625  D''.     One-tenth  of 
the  thickness  of  tooth  at  pitch-line  is  more  than  one- 
sixteenth  of  working  depth,  being  .07854  D '. 

Example.  Example.— Wheel    30   teeth,    If   circular    pitch. 

sizes  of  Blank  P'  =  l-6"  ;  then  £=.75''  or  thickness  of  tooth  equals  £". 
Partds  fo^Gelr  «  =  l-5''X.3183  =  .4775"  =  a  diameter  pitch.  (See 
of  30  teeth,  itf  tables  of  tooth  parts,  pages  68-71). 

in. circular 

pitch.  D'=30x.4775"  =  14.325"  =  diameter  of  pitch-circle, 

D=(30+2)X.4775"  =  15.280"  =  diameter  of  adden- 
dum circle. 

/=TV  of  .75'' =.075"  =  clearance  at  bottom  of  space, 

D"  =  2x.4775"  =  .9549"=working  depth  of  teeth. 

D"  +  /^2x.4775"  +  .075"  =  1.0299"=whole  depth  of 
space. 

«+/=. 4775"  +  . 075"  =  . 5525"=depth  of  space  inside 
of  pitch-line. 

~D"=2s  or  the  working  depth  of  teeth  is  equal  to 
two  diameter  pitches. 

In  making  calculations  it  is  well  to  retain  the  fourth 
place  in  the  decimals,  but  when  drawings  are  passed 
into  the  workshop,  three  places  of  decimals  are  suffi- 
cient. 


PROVIDENCE,    R.    I. 


1IT 


FIG.  5,  SPUR  GEARING. 


8  BKOWN   &   SHAKPE   MFG.    CO. 


iw^utanceente?s     The  ^s^1106  between  the  centers  of  two  wheels  is 

of  two  Gears,  evidently  equal  to  the  radius  of  pitch-circle  of  one  wheel 

added  to  that  of  the  other.     The  radius  of  pitch-circle 

is  equal  to  s  multiplied  by  one-half  the  number  of  teeth 

in  the  wheel. 

Hence,  if  we  know  the  number  of  teeth  in  two  wheels, 
in  mesh,  and  the  circular  pitch,  to  obtain  the  distance 
between  centers  we  first  find  s  ;  then  multiply  s  by  one- 
half  the  sum  of  number  of  teeth  in  both  wheels  and  the 
product  will  be  distance  between  centers. 

Example  —  What  is  the  distance  between  the  centers 
of  two  wheels  35  and  60  teeth,  1  J"  circular  pitch.  We 
first  find  s  to  be  lj"x  .3183  =  .3979".  Multiplying  by 
47.5  (one-half  the  sum  of  35  and  60  teeth)  we  obtain 
18.899"  as  the  distance  between  centers. 

Shr\°nTan|e  fhi  Pattem  Gears  should  be  made  large  enough  to 
Gear  castings,  allow  for  shrinkage  in  casting.  In  cast-iron  the  shrinkage 
is  about  J  inch  in  one  foot.  For  gears  one  to  two  feet 
in  diameter  it  is  well  enough  to  add  simply  -j-J-g-  of 
diameter  of  finished  gear  to  the  pattern.  In  gears 
about  six  inches  diameter  or  less,  the  moulder  will 
generally  rap  the  pattern  in  the  sand  enough  to  make 
any  allowance  for  shrinkage  unnecessary.  In  pattern 
gears  the  spaces  between  teeth  should  be  cut  wider 
than  finished  gear  spaces  to  allow  for  rapping  and  to 
avoid  having  too  much  cleaning  to  do  in  order  to  have 
gears  run  freely.  In  cut  patterns  of  iron  it  is  generally 
Metal  Pattern  enough  to  make  spaces  .015"  to  .02"  wider.  This 
makes  clearance  .03"  to  .04"  in  the  patterns.  Some 
moulders  might  want  .06"  to  .07"  clearance. 

Metal  patterns  should  be  cut  straight  ;  they  work 
better  with  no  draft.  It  is  well  to  leave  about  .005"  to 
be  finished  from  side  of  patterns  after  teeth  are  cut  ; 
this  extra  stock  to  be  taken  away  from  side  where 
cutter  comes  through  so  as  to  take  out  places  wiiere 
fctock  is  broken  out.  The  finishing  should  be  done 
with  file  or  emery  wheel,  as  turning  in  a  lathe  is  likely 
to  break  out  stock  as  badly  as  a  cutter  might  do. 

If  cutters  are  kept  sharp  and  care  is  taken  when 
coming  through  the  allowance  for  finishing  is  not  nec- 
essary and  the  blanks  may  be  finished  before  they  are 
cut. 


PROVIDENCE,    K.    I. 


CHAPTER   III. 
SINGLE-CURVE  GEARS  OF  30  TEETH  ABD  OVER, 


Single-curve  teeth  are  so  called  because  they  have  T^f6  Curve 
but  one  curve  by  theory,  this  curve  forming  both  face 
and  flank  of  tooth  sides.  In  any  gear  of  thirty  teeth 
and  more,  this  curve  can  be  a  single  arc  of  a  circle 
whose  radius  is  one-fourth  the  radius  of  the  pitch 
circle.  In  gears  of  thirty  teeth  and  more,  a  fillet  is 
added  at  bottom  of  tooth,  to  make  it  stronger,  equal 
in  radius  to  one-sixth  the  widest  part  of  tooth  space. 

A  cutter  formed  to  leave  this  fillet  has  the  advantage 
of  wearing  longer  than  it  would  if  brought  up  to  a 
corner, 

In  gears  less  than  thirty  teeth  this  fillet  is  made  the 
same  as  just  given,  and  sides  of  teeth  are  formed  with 
more  than  one  arc,  as  will  be  shown  in  Chapter  VI. 

Having  calculated  the  data  of  a  gear  of  30  teeth,  £    Example  of  a 
inch  circular  pitch  (as  we  did  in  Chapter  II.  for  1£"  =%"' 
pitch),  we  proceed  as  follows  : 

1.  Draw  pitch  circle  and  point  it  off  into  parts  equal     Geometrical 

T     ,P  ,,          .         T  .     '  Construction. 

to  one-halt  the  circular  pitch.  Fig.  6. 

2.  From  one  of  these  points,  as  at  B,  Fig.  6,  draw 
radius  to  pitch  circle,  and  upon  this  radius  describe  a 
semicircle ;  the  diameter  of  this  semicircle  being  equal 
to  radius  of  pitch  circle.      Draw  addendum,  working 
depth  and  whole  depth  circles. 

3.  From  the  point  B,  Fig.  6,  where  semicircle,  pitch 
circle  and  outer  end  of  radius  to  pitch  circle  meet,  lay 
off  a  distance  upon  semicircle  equal  to  one-fourth  the 
radius  of  pitch  circle,  shown  in  the  figure  at  BA,  and 
is  laid  off  as  a  chord. 

4.  Through  this  new  point  at  A,  upon  the  semicircle, 
draw  a  circle  concentric  to  pitch  circle.     This  last  is 


10 


BROWN    &    SHAEPE    MFG.    CO. 


.  6 


GEAR,  30  TEETH, 
f'CIRCULAR  PITCH 
P'=for  .75' 
N=30 

P  =4.1888' 
t  =  .375' 
S  =  .2387" 
D"-=  .4775' 
8-f-/=  .2762' 
.5150' 

D'=7.1610' 
D  =7.6384' 


SINGLE   CURVE   GEAR. 


PEOVIDENCE,    R.    I.  11 

called  the  base  circle,  and  is  the  one  for  centers  of 
tooth  arcs.  In  the  system  of  single  curve  gears,  we 
have  adopted  the  diameter  of  this  circle  is  .968  of  the 
diameter  of  pitch  circle.  Thus  the  base  circle  of  any 
gear  1  inch  pitch  diameter  by  this  system  is  .968". 
If  the  pitch  circle  is  2"  the  base  circle  will  be  1.936." 

5.  With  dividers  set  to  one-quarter  of  the  radius  of 
pitch  circle,  draw  arcs  forming  sides  of  teeth,  placing 
one  leg  of  the  dividers  in  the  base  circle  and  letting 
the  other  leg  describe  an  arc  through  a  point  in  the 
pitch  circle  that  was  made  in  laying  off  the  parts  equal 
to  one-half  the  circular  pitch.     Thus  an  arc  is  drawn 
about  A  as  center  through  B. 

6.  With  dividers  set  to  one-sixth  of  the  widest  part 
of  tooth  space,  draw  the  fillets  for  strengthening  teeth 
at  their  roots.     These  fillet  arcs  should  just  touch  the 
whole   depth   circle   and   the   sides   of   teeth   already 
described. 

Single  curve  or  involute  gears  are  the  only  gears  Invomte^Gea?- 
that  can  run  at  varying  distance  of  axes  and  transmit ing< 
unvarying  angular  velocity.     This  peculiarity  makes 
involute  gears   specially  valuable  for  driving  rolls  or 
any  rotating   pieces,  the   distance  of  whose   axes   is 
likely  to  be  changed. 

The  assertion  that  gears  crowd  harder  on  bearings    Pressure  on 

bearings. 

when  of  involute  than  when  of  other  forms  of  teeth, 
has  not  been  proved  in  actual  practice. 

Before  taking-  next  chapter,  the  learner  should  make    Practice,  be- 

i     -,          •  ,  '      ,       ,,  n  •       a       fore    taking 

several  drawings   of  gears  30  teeth  and  more.     Say  next  chapter. 
make  35  and  70  teeth  !£"  P'.     Then  make  40  and  65 
teeth  I"  P'. 

An  excellent  practice  will  be  to  make  drawing  on 
cardboard  or  Bristol-board  and  cut  teeth  to  lines,  thus 
making  paper  gears ;  or,  what  is  still  better,  make  them 
of  sheet  metal.  By  placing  these  in  mesh  the  learner 
can  test  the  accuracy  of  his  work. 


12  BKOWN   &    SHAKPE   MFQ.    CO. 


CHAPTER    IV. 

RACK  TO  MESH  WITH  SINGLE-CURVE  GEARS  HAVING 
30  TEETH  AND  OVER. 


madeaprepam-     ^is  £ear  (^S-  "0  *s  made  precisely  the  same  as  gear 
Xckdrawins  in  Cnapter  HI.    It  makes  no  difference  in  which  direc- 
tion the  construction  radius  is  drawn,  so  far  as  obtain- 
ing form  of  teeth  and  making  gear  are  concerned. 

Here  the  radius  is  drawn  perpendicular  to  pitch  line 
of  rack  and  through  one  of  the  tooth  sides,  B.  A  semi- 
circle is  drawn  on  each  side  of  the  radius  of  the  pitch 
circle. 

The  points  A  and  A'  are  each  distant  from  the  point 
B,  equal  to  one-fourth  the  radius  of  pitch  circle  and 
correspond  to  the  point  A  in  Fig.  6. 

In  Fig.  7  add  two  lines,  one  passing  through  B  and 
A  and  one  through  B  and  A'.  These  two  lines  form 
angles  of  75^°  (degrees)  with  radius  BO.  Lines  BA 
and  BA'  are  called,  lines  of  pressure.  The  sides  of 
rack  teeth  are  made  perpendicular  to  these  lines. 
Rack.  A  Rack  is  a  straight  piece,  having  teeth  to  mesh 

with  a  gear.  A  rack  may  be  considered  as  a  gear  of 
infinitely  long  radius.  The  circumference  of  a  circle 
approaches  a  straight  line  as  the  radius  increases, 'and 
when  the  radius  is  infinitely  long  any  finite  part  of  the 
construction  circumference  is  a  straight  line.  The  pitch  line  of  a 

of  Pitch  Line  of 

Rack.  rack,  then,  is  merely  a  straight  line  just  touching  the 

pitch  circle  of  a  gear  meshing  with  the  rack.      The 

thickness    of    teeth,    addendum    and  depth   of    teeth 

below  pitch  line  are  calculated  the  same  as  for  a  wheel. 

(For  pitches  in  common  use,  see  table  of  tooth  parts.) 

The  term  circular  pitch  when  applied  to  racks  can  be 

more  accurately  replaced   by   the   term   linear  pitch 

Linear  applies  strictly  to  a  line  in  general  while  circular 

pertains  to  a  circle.     Linear  pitch  means  the  distance 

between  the  centres    of  two   teeth  on   the  pitch  line 

whether  the  line  is  straight  or  curved. 


PROVIDENCE,    R.    I. 

A  rack  to  mesh  with  a  single-curve  gear  of  30  teeth 
or  more  is  drawn  as  follows : 

1.  Draw  straight  pitch  line  of  rack ;  also  draw  ad- 
dendum line,  working  depth  line  and  whole  depth  line, 
each  parallel  to  the  pitch  line  (see  Fig.  7). 


Rack. 
Fig.  7. 


RACK  TO  MESH  WITH  SINGLE  CURVE  GEAR 
HAVING  30  TEETH  AND  OVER. 


14  BROWN    &    SHARPE    MFG.    CO. 

2.  Point  off  the  pitch  line  into  parts  equal  to  one- 
half  the  circular  pitch,  or  =  t. 

3.  Through  these  points  draw  lines  at  an  angle  of 
75£°  with  pitch  lines,  alternate  lines  slanting  in  oppo- 
site directions.     The  left-hand  side  of  each  rack  tooth 
is  perpendicular  to  the  line  BA.     The  right-hand  side 
of  each  rack  tooth  is  perpendicular  to  the  line  BA'. 

4.  Add  fillets  at  bottom  of  teeth  equal  to  \  of  the 
width  of  spaces  between  the  rack  teeth  at  the  adden- 
dum line. 

side?Sofe  nick      Ttie  sketch,  Fig.  8,  will  show  how  to  obtain  angle  cf 
Teeth.  sides  of  rack  teeth,  directly  from  pitch  line  of  rack, 

without  drawing  a  gear  in  mesh  with  the  rack. 


Upon  the  pitch  line  b  b',  draw  any  semicircle — 
baa'  b'.  From  point  b  lay  off  upon  the  semicircle 
the  distance  b  a,  equal  to  one-quarter  of  the  diameter 
of  semicircle,  and  draw  a  straight  line  through  b  and  a. 

This  line,  b  a,  makes  an  angle  of  75 \°  with  pitch  line 
bb',  and  can  be  one  side  of  rack  tooth.  The  same 
construction,  b'  a',  will  give  the  inclination  75£°  in  the 
opposite  direction  for  the  other  side  of  tooth. 

The  sketch,  Fig.  9,  gives  the  angle  of  sides  of  a  tool 
for  planing  out  spaces  between  rack  teeth.  Upon  any 
line  OB  draw  circle  OABA'.  From  B  lay  off  distance 
BA  and  BA',  each  equal  to  one-quarter  of  diameter  of 
the  circle. 

Draw  lines  OA  and  OA'.  These  two  lines  form  an 
angle  of  29°,  and  are  right  for  inclination  of  sides  of 
rack  tool. 


PROVIDENCE,    R.    I. 

Make  end  of  rack  tool  .31  of  circular  pitch,  and  then 
round  the  corners  of  the  tool  to  leave  fillets  at  the 
bottom  of  rack  teeth. 

Thus,  if  the  circular  pitch  of  a  rack  is  1J"  and  we 
multiply  by  .31,  the  product  .465"  will  be  the  width  of 
tool  at  end  for  rack  of  this  pitch  before  corners  are 
taken  off.  This  width  is  shown  at  x  y. 


This  sketch  and  the  foregoing  rule  are  also  right  for  worm  Thread 
a   worm-thread  tool,  but    a  worm-thread  tool  is  not 
usually  rounded  for  fillet.     In  cutting  worms,  leave 
width   of   top  of   thread  .335    of   the    circular   pitch. 
When  this  is  done,  the  depth  of  thread  will  be  right. 


16  BROWN    &    SHAKPE    MFQ.    CO. 


CHAPTER  V. 

DIAMETRAL  PITCH— SIZING  BLANKS  AND  TEETH— DISTANCE 
BETWEEN  CENTRES  OF  WHEELS. 


In  making  drawings  of  gears,  and  in  cutting  racksr 

necessary  to  it  is  necessary  to  know  the  circular  pitch,  both  on 

cuiar  Pitch.      account  of  spacing  teeth  and  calculating  their  strength. 

It  would  be  more  convenient  to  express  the  circular 

pitch  in  whole  inches,  and  the  most  natural  divisions 

in  a  complete  of  an  inch,  as  1"  P',  f"  P',  y  P',  and  so  on.     But  as 

Pitch  ecircum!  the  circumference  of  the  pitch  circle  must  contain  the 


-  circular    pitch  some  whole  number  of  times,    corre- 
whofe  spending   to   the   number   of   teeth   in   the  gear,  the 


Smes.ber  ot  diameter  of  the  pitch  circle  will  often  be  of  a  size  not 
readily  measured  with  a  common  rule.  This  is  because 
the  circumference  of  a  circle  is  equal  to  3.1416  times 
the  diameter,  or  the  diameter  is  equal  to  the  circum- 
ference multiplied  by  .3183. 

In  practice,  it  is  better  that  the  diameter  should  be 
Pitch,  i  n  of  some  size  conveniently  measured.    The  same  applies 

Terms   of    the  J 

Diameter.  to  the  distance  between  centers.  Hence  it  is  generally 
more  convenient  to  assume  the  pitch  in  terms  of  the 
diameter.  In  Chapter  II.  was  given  a  definition  of  a- 
diameter  pitch,  and  also  how  to  get  a  diameter  pitch 
from  the  circular  pitch. 

We  can  also  assume  a  diameter  pitch  and  pass  to  its 
circular  Pitch  equivalent  circular  pitch.     If  the  circumf  erence  of  the 

ter  Pitch.  pitch  circle  is  divided  by  the  number  of  teeth  in  the 
gear,  the  quotient  will  be  the  circular  pitch.  In  the 
same  manner,  if  the  diameter  of  the  pitch  circle  is 
divided  by  the  number  of  teeth,  the  quotient  will  be  a 
diameter  pitch.  Thus,  if  a  gear  is  12  inches  pitch 
diameter  and  has  48  teeth,  dividing  12"  by  48,  the 
quotient  £  "  is  a  diameter  pitch  of  this  gear.  In  prac- 


PROVIDENCE,    R.    I.  17 

tice,  a  diameter  pitch  is  taken  in  some  convenient  part 

of    an   inch,    as    i"  diameter  pitch,  and  so    on.      It    Abbreviation 
a  of  Diameter 

is  convenient  in  calculation  to  designate  one  of  these  pitcl1- 
diameter  pitches  by  s,  as  in  Chapter  II.  Thus,  for  J" 
diameter  pitch,  s  is  equal  to  J".  Generally,  in  speak- 
ing of  diameter  pitch,  the  denominator  of  the  fraction 
only  is  named.  J"  diameter  pitch  is  then  called  3 
diametral  pitch.  That  is,  it  has  been  found  more  con- 
venient to  take  the  reciprocal  of  a  diameter  pitch  in 
making  calculation.  The  reciprocal  of  a  number  is  1, 
divided  by  that  number.  Thus  the  reciprocal  of  J 
4,  because  ^  goes  into  1  four  times. 

Hence,  we  come  to  the  common  definition  : 

DIAMETRAL  PITCH  is  the  number  of.  teeth  to  one  inch  pitdtx.me 
of  diameter  of  pitch  circle.  Let  this  be  denoted  by  P. 
Thus,  J"  diameter  pitch  we  would  call  4  diametral 
pitch  or  4  P,  because  there  would  be  4  teeth  to  every 
inch  in  the  diameter  of  pitch  circle.  The  circular 
pitch  and  the  different  parts  of  the  teeth  are  derived 
from  the  diametral  pitch  as  follows. 

s-^is  =  P',  or  3.1416  divided  by  the  diametral  pitch  J^^6^ 
is  equal  to  the  circular  pitch.     Thus  to  obtain  the  cir-t^  Circular 
cular  for  4  diametral  pitch,  we  divide  3.1416  by  4  and 
get  .7854  for  'the  circular  pitch,  corresponding  to  4c£^ta^nitcc1£ 
diametral  pitch.  j£Xpi£Lame" 

In  this  case  we  would  write  P=4,P'=.7854",  s=J". 
^"—Sy  or  one  inch  divided  by  the  number  of  teeth  to 
an  inch,  gives  distance  on  diameter  of  pitch  circle 
occupied  by  one  tooth.  The  addendum  or  face  of 
tooth  is  the  same  distance  as  s. 

|=P,  or  one  inch  divided  by  the  distance  occupied 
by  one  tooth  equals  number  of  teeth  to  one  inch. 

1-'f3=«,  or  1.57  divided  by  the  diametral  pitch  gives 
thickness  of  tooth  at  pitch  line.  Thus,  thickness  of 
teeth  along  the  pitch  line  for  4  diametral  pitch  is  .392".  gj;he  Pitch 

£=D',  or  number  of  teeth  in  a  gear  divided  by  Uw- 
diametral  pitch  equals  diameter  of  the  pitch  circle. 
Thus  for  a  wheel,  60  teeth,  12  P,  the  diameter  ot 
pitch  circle  will  be  5  inches.  g^'  Pitch 

Z=D,  or  add  2  to  the  number  of  teeth  in  a  wheel  N  umber   o? 


and  divide  the  sum  by  the  diametral  pitch,  and 

tral  Pitch  to 
find  the  Wholo 
Diameter. 


18  BROWN   &    SHAEPE   MFG.    CO. 

quotient  will  be  the  whole  diameter  of  the  gear  or  the 
diameter  of  the  addendum  circle.  Thus,  for  60  teeth, 
12  P,  the  diameter  of  gear  blank  will  be  5fy  inches. 

D,=P,  or  number  of  teeth  divided  by  diameter  of 
pitch  circle  in  inches,  gives  the  diametral  pitch  or 
number  of  teeth  to  one  inch.  Thus,  in  a  wheel,  24 
teeth,  3  inches  pitch  diameter,  the  diametral  pitch  is  8. 

^~=Pt  or  add  2  to  the  number  of  teeth;  divide  the 
sum  by  the  whole  diameter  of  gear,  and  the  quotient 
will  be  the  diametral  pitch.  Thus,  for  a  wheel  3^" 
diameter,  14  teeth,  the  diametral  pitch  is  5. 

P  D'=N,  or  diameter  of  pitch  circle,  multiplied  by 
diametral  pitch  equals  number  of  teeth  in  the  gear. 
Thus,  in  a  gear,  5  pitch,  8"  pitch  diameter,  the  number 
of  teeth  is  40. 

—  =  s,  or  divide  the  whole  diameter  of  a  spur  gear 
by  the  number  of  teeth  plus  two,  and  the  quotient 
will  be  the  addendum,  or  a  diameter  pitch. 

FitchDiametei  IQ  future?  when  we  speak  of  a  diameter  pitch,  we 
shall  mean  the  addendum  distance  or  s.  If  we  speak 
of  so  many  diameter  pitches,  we  shall  mean  so  many 

raihKtc£iame~  ^mes  s>  (7  =  s)*  When  we  say  the  diametral  pitch  we 
shall  mean  the  number  of  teeth  to  one  inch  of  diameter 
of  pitch  circle,  or  P,  (j=P). 

ametSSi^Sh     When  the  circular  pitch  is  given,  to  find  the  corre- 

Fttch.  circular  spending  diametral  pitch,  divide  3.1416  by  the  circular 
pitch.  Thus  1.57  P  is  the  diametral  pitch  correspond- 
ing to  2-inch  circular  pitch,  (—£^='P). 

Example.  What  diametral  pitch  corresponds  to  J"  circular 

pitch  ?  Remembering  that  to  divide  by  a  fraction  we 
multiply  by  the  denominator  and  divide  by  the  numer- 
ator, we  obtain  6.28  as  the  quotient  of  3.1416  divided  by 
J  .  6.28  P,  then,  is  the  diametral  pitch  corresponding 
to  ^  circular  pitch.  This  means  that  in  a  gear  of  $ 
inch  circular  pitch  there  are  six  and  twenty-eight  one 
hundredths  teeth  to  every  inch  in  the  diameter  of  the 
pitch  circle.  In  the  table  of  tooth  parts  the  diametral 
pitches  corresponding  to  circular  pitches  are  carried 
out  to  four  places  of  decimals,  but  in  practice  two 
places  of  decimals  are  enough. 


PROVIDENCE,    B.    I. 

When  two  gears  are.  in  niesh,  so  that  their  pitch 
circles  just  touch,  the  distance  between  their  axes  or 
centers  is  equal  to  the  sum  of  the  radii  of  the  two 
gears.  The  number  of  the  diameter  pitches  between 
centers  is  equal  to  half  the  sum  of  number  of  teeth  in 
both  gears.  This  principle  is  the  same  as  given  in 
Chapter  II.,  page  6,  but  when  the  diametral  pitch  and  D?£[£CJ0  fl^ 
numbers  of  teeth  in  two  gears  are  given,  add  together  tween  centers. 
the  numbers  of  teeth  in  the  two  wheels  and  divide  half 
the  sum  by  the  diametral  pitch.  The  quotient  is  the 
center  distance. 

A  gear  of  20  teeth,  4  P,  meshes  with  a  gear  of  50  Example. 
teeth:  what    is  the  distance  between  their   axes    or 
centers  ?     Adding  50  to  20  and  dividing  half  the  sum 
by  4,  we  obtain  8f "  as  the  center  distance. 

The  term  diametral  pitch  is  also  applied  to  a  rack. 
Thus,  a  rack  3  P,  means  a  rack  that  will  mesh  with  a 
gear  of  3  diametral  pitch. 

It  will  be  seen  that  if  the  expression  for  a  diameter     Fractional 

..    ,     ,  n  D  i  a  in  e  t  ral 

pitch  has  any  number  except  1  for  a  numerator,  we  Pitch, 
cannot  express  the  diametral   pitch  by  naming   the 
denominator  only.     Thus,  if  the  addendum  or  a  diam- 
eter pitch  is  ^,  the  diametral  pitch  will  be  2£,  because 
1  divided  by  -fo  equals  2£. 


20  BKOWN   &    SiUEPE    MFG.    CO. 


CHAPTER     VI. 

SINGLE-GDRYE  GEARS  HAYING  LESS  THAN  30  TEETH— GEARS  AND 
RACKS  TO  MESH  WITH  GEARS  HAYING  LESS  THAN  30  TEETH. 


Construction,  jn  j^g,  iQ,  the  construction  of  the  rack  is  the  same 
as  the  construction  of  the  rack  in  Chapter  IV.  The 
gear  in  Fig.  10  is  drawn  from  base  circle  out  to  adden- 
dum circle,  by  the  same  method  as  the  gear  in  Chapter 
III.,  but  the  spaces  inside  of  base  circle  are  drawn  as 
follows  : 

G^rs^iow     In   gears>    12   and    13   teeth>    tlie   sides   of   spaces 

TeeSiber9   ofmside  of  base  circle  are  parallel  for  a  distance  not 

more  than  J  of  a  diameter  pitch,  or  J  s  ;    gears  14, 

15  and  16  teeth  not  more  than  -J-  s  ;   17  to  20  teeth, 

not  more  than  -J-  s.     In  gears  with  more  than  20  teeth 

the  parallel  construction  is  omitted. 

construction      Then,  with   one   leg  of  dividers  in   pitch  circle  in 

of  Fig.  10  con-  '  * 


center  of  next  tooth,  e,  and  other  leg  just  touching 
one  of  the  parallel  lines  at  b,  continue  the  tooth  side 
into  c,  until  it  will  touch  a  fillet  arc,  whose  radius  is 
-J-  the  width  of  space  at  the  addendum  circle.  The 
part,  b'  c',  is  an  arc  from  center  of  tooth  g,  etc.  The 
flanks  of  teeth  or  spaces  in  gear,  Fig.  11,  are  made  the 
same  as  those  in  Fig.  10. 

This  rule  is  merely  conventional  or  not  founded 
upon  any  principle  other  than  the  judgment  of  the  de- 
signer, to  effect  the  object  to  have  spaces  as  wide  as 
practicable,  just  below  or  inside  of  base  circle,  and 
then  strengthen  flank  with  as  large  a  fillet  as  will  clear 
addenda  of  any  gear.  If  flanks  in  any  gear  will  clear 
addenda  of  a  rack,  they  will  clear  addenda  of  any 
Internal  Gear,  other  gear,  except  internal  gears.  An  internal  gear  is 
one  having  teeth  upon  the  inner  side  of  a  rim  or  ring. 
Now,  it  will  be  seen  that  the  gear,  Fig.  10,  has  teeth 


UNIVERSITY 


PKOVIDENCE,    E.    I. 


21 


22  BEOWN    &    SHARPE   MFG.    CO. 

too  much  rounded  at  the  points  or  at  the  addendum 
circle.     In  gears  of  pitch  coarser  than  10  to  inch  (10 
AddTndnf  of  I>)>   ancl    having   less   than  30   teeth,    this   rounding 
Teeth.  becomes  objectionable.    This  rounding  occurs,  because 

in  these  gears  arcs  of  circles  depart  too  far  from  the 
true    involute  curve,   being  so  much  that  points   of 
teeth  get  no  bearing  on  flanks  of  teeth  in  other  wheels. 
In  gear,  Fig.  11,  the  teeth  outside  of  base  circle  are 
made  as  nearly  true  involute  as  a  workman  will  be  able 
to  get  without  special  machinery.  This  is  accomplished 
tionPtorTmein- as  f °^ows :  draw  three  or  four  tangents  to  the  base 
volute.  circle,  i  i',  j  j',  k  k',  II',  letting  the  points  of  tangency 

on  base  circle  i',j',  k',  I'  be  about  \  or  J  the  circular  pitch 
apart ;  the  first  point,  i\  being  distant  from  *',  equal  to 
J  the  radius  of  pitch  circle.  With  dividers  set  to  J 
the  radius  of  pitch  circle,  placing  one  leg  in  i',  draw 
the  arc,  a'  i  j;  with  one  leg  in  j',  and  radius  j'  j, 
diawj  k;  with  one  leg  in  k',  and  radius  k'  k  draw  k  L 
Should  the  addendum  circle  be  outside  of  Z,  the  tooth 
side  can  be  completed  with  the  last  radius,  I'  I.  The 
arcs,  a'  ij,  j  k  and  k  I,  together  form  a  very  close 
approximation  to  a  true  involute  from  the  base  circle, 
i'  jf  k'  I'.  The  exact  involute  for  gear  teeth  is  the 
curve  made  by  the  end  of  a  band  when  unwound  from 
a  cylinder  of  the  same  diameter  as  base  circle. 

The   foregoing    operation  of   drawing  tooth  sides, 
although  tedious  in  description,  is  very  easy  of  practical 
application. 
Hounding   of     It  will  also  be  seen  that  the  addenda  of  rack  teeth 

Adde  nda  of 

Rack.  in  Fig.  10,  interfere  with  the  gear-teeth  flanks,  as  at 

in  n;  to  avoid  this  interference,  the  teeth  of  rack,  Fig. 
11,  are  rounded  at  points  or  addenda. 

It  is  also  necessary  to  round  off  the  points  of  invo- 
lute teeth  in  high -numbered  gears,  when  they  are  to 
interchange  with  low-numbered  gears.  In  interchange- 
able sets  of  gears  the  lowest-numbered  pinion  is  usual- 

Tempietsly  12.     Just  how  much  to  round  off  is  best  learnt  by 
necessary  for  * 

Rounding  off  makincf  templets  of   a  few  teeth  out  of  thin  metal   or 

Points  of  teeth.  fo  /• 

cardboard,  tor  the  gear  and  rack,  or,  two  gears  re- 
quired, and  fitting  addenda  of  teeth  to  clear  flanks. 
However  accurate  we  may  make  a  diagram,  it  is  quite 


PKOVIDENCE,    K.    I. 


23 


SINGLE  CURVE  GEAR,  2  P.,  12  TEETH. 
IN  MESH  WITH  RACK. 

P  =2 
N  =12 
P'=   1.57' 
t  =     .7854' 
S  =     .500" 
D '—  1.000' 
s+/=    .5785' 

=  1.078' 
D'=6.' 


24  BROWN    &    SHARPE    MFG.    CO. 

as  well   to  make  templets  in  order  to  shape  cutters 
accurately 

a  ^etgro?8cu£      ^  *s  kest  to  make  cutters  to  corrected  diagrams,  as 
ters-  in  Fig.  11.     When  corrected  diagrams  are  made,  as 

in  Fig.  1 1 ,  take  the  following  : 

For  12  and  13  teeth,  diagram  of  12  teeth. 
"     14    to    16       "  «         "   14       " 

t<      17     k'     20       "  "         "   17       " 

"     21     l*     25       u  "         "  21       u 

"     26     u     34       "  "         "  26       " 

u     35     "     54       <«  "         "  35       " 

u     55     "   134       «<  4<         "  55       " 

«'  135    4<rack,     u  "         "135       " 

Templets  for  large  gears  must  be  fitted  to  run  with 
12  teeth,  etc. 


PROVIDENCE,  R.  I. 


25 


CHAPTER  VII. 
DODBLE-CDRYE  TEETH— GEAR,  15  TEETH— RACK, 


In  double-curve  teeth  the  formation  of  tooth  sides 
changes  at  the  pitch  line.  In  all  gears  the  part  of  Faces  are  Con- 
teeth  outside  of  pitch  line  is  convex ;  in  some  gears 
the  sides  of  teeth  inside  pitch  line  are  convex ;  in  some, 
radial ;  in  others,  concave.  Convex  faces  and  concave 
flanks  are  most  familiar  to  mechanics.  In  interchange- 
able sets  of  gears,  one  gear  in  each  set,  or  of  each 
pitch,  has  radial  flanks.  In  the  best  practice,  this  gear 
has  fifteen  teeth.  Gears  with  more  than  fifteen  teeth, 
have  concave  flanks ;  gears  with  less  than  fifteen  teeth, 
have  convex  flanks.  Fifteen  teeth  is  called  the  Base 
•of  this  system. 

We  will  first  draw  a  gear  of  fifteen  teeth.      This  of  construction 
fifteen-tooth   construction   enters   into   gears  of   any 
number  of  teeth  and  also  into  racks.     Let  the  gear  be 
3  P.     Having  obtained  data,  we  proceed  as  follows : 

1.  Draw  pitch  circle  and  point  it  oft  into  parts  equal 
to  one-thirtieth  of  the  circumference,  or  equal  to  thick- 
ness of  tooth  —  t. 

2.  From  the  center,  through  one  of  these  points,  as 
at  T,  Fig.  12,  draw  line  OTA.     Draw  addendum  and 
whole-depth  circles. 

3.  About  this  point,  T,  with  same  radius  as  15- tooth 
pitch  circle,  describe  arcs  A  K  and  O  k.    For  any  other 
double-curve  gear  of  3  P.,  the  radius  of  arcs,  A  K  and 
O  k,  will  be  the  same  as  in  this  15-tooth  gear=2J*. 
In  a  15-tooth  gear,  the  arc,  O  &,  passes  through  the 
-center  O,  but  for  a  gear  having  any  other  number  of 
teeth,   this  construction   arc  does   not   pass   through 
center  of  gear.    Of  course,  the  15-tooth  radius  of  arcs, 
A  K  and  O  k,  is  always  taken  from  the  pitch  we  are 
working  with. 


26 


BROWN   &    SHARPE   MFG.    CO. 


G~EAR,3  P.,  15  TEETH 

P*-3 
N— 15 
P'=  1..Q472" 
t  =   .5236' 
8—   .3333' 
D'fe-   .6666' 
«+f  -    .3857' 
D"+/=    .7190' 
D'=^  5.0000' 
D  =5,6666' 


w 


Fig.  13 

DOUBLE    CURVE    GEAR. 


PROVIDENCE,    R.    I.  27 

4.  Upon  these  arcs  on  opposite  sides  of  lines  OTA, 
lay  off  tooth  thickness,  A  K  and  O  &,  and  draw  line 
KT  k. 

5.  Perpendicular  to  K  T  k,  draw  line  of  pressure, 
L  T  P ;  also  through  O  and  A,  draw  lines  A  E  and  O  r, 
perpendicular  to  K  T  k.    The  line  of  pressure  is  at 
an  angle  of  78°  with  the  radius  of  gear. 

6.  From  O,  draw  a  line  O  K  to  intersection  of  A  E 
with  K  T  &.     Through  point  c,  where  O  E  intersects 
L  P,  describe  a  circle  about  the  center,  O.     In  this 
circle  one  leg  of  dividers  is  placed  to  describe  tooth 
faces 

7.  The  radius,    c  d,   of  arc  of  tooth   faces   is   the 
straight  distance  from  c  to  tooth-thickness  point,  5, 
on  the  other  side  of  radius,  O  T.     With  this  radius,  c  b, 
describe  both  sides  of  tooth  faces. 

8.  Draw  flanks  of  all  teeth  radial,  as  O  e  and  O  /. 
The  base  gear,  15  teeth  only,  has  radial  flanks. 

9.  With  radius  equal  to  one  sixth  of  the  widest  part 
of  space,  as  g  h,  draw  fillets  at  bottom  of  teeth. 

The  foregoing  is  a  close  approximation  to  epicy-      Approxima- 

tion    to    Epicy- 

cloidal  teeth.  To  get  exact  teeth,  make  two  15- too  theioidai  Teeth, 
gears  of  thin  metal.  Make  addenda  long  enough  to 
come  to  a  point,  as  at  n  and  q.  Make  radial  flanks,  as 
at  m  and  p,  deep  enough  to  clear  addenda  when  gears 
are  in  mesh.  First  finish  the  flanks,  then  fit  the  long 
addenda  to  the  flanks  when  gears  are  in  mesh. 

When  these  two  templet  gears  are  alike,  the  center S 
are  the  right  distance  apart  and  the  teeth  interlock 
without  backlash,  they  are  exact.  One  of  these  tem- 
plet gears  can  now  be  used  to  test  any  other  templet 
gear  of  the  same  pitch. 

Gears  and  racks  will  be  right  when  they  run  cor- 
rectly with  one  of  these  15-tooth  templet  gears.  Five 
or  six  teeth  are  enough  to  make  in  a  gear  templet. 

DOUBLE-CURVE   EACK. — Let   us  draw   a  rack    3    P. ,, Double-curve 

xtaCK,  r  Ig.  lo. 

Having  obtained  data  of  teeth  we  proceed  as  follows : 

1.  Draw  pitch  line  and  point  it  off  in  parts  equal 
to  one- half  the  circular  pitch.     Draw  addendum  and 
whole-depth  lines. 

2.  Through  one  of  the  points,  as  at  T,  Fig.  13,  draw 
line  OTA  perpendicular  to  pitch  line  of  rack. 


28 


BROWN    &    SHARPE    MFG.    CO. 


RACK,  3  P. 
P=  3 

p/=  1.0472" 
t  =  .5236' 
8  =  .3333' 
D'=  .6666' 
8+/=  .3857" 


.7190' 


.  13 

DOUBLE  CURVE    RACK, 


PROVIDENCE,    R.    I. 

3.  About  T  make  precisely  the  same  construction  as 
was  made  about  T  in  Fig.  12.     That  is,  with  radius  of 
15-tooth  pitch  circle  and  center  T  draw  arcs  O  k  and 
A  K ;  make   O  k  and  A  K  equal  to  tooth  thickness ; 
draw  K  T  k  •  draw  O  r,  A  R,  and  line  of  pressure,  each 
perpendicular  to  K  T  k. 

4.  Through  E.  and  r,  draw  lines  parallel  to   O  A. 
Through  intersections  c  and  c'  of  these  lines,  with 
pressure  line  L  P,  draw  lines  parallel  to  pitch  line. 

5.  In  these  last  lines  place  leg  of  dividers,  and  draw 
faces  and  flanks  of  teeth  as  in  sketch. 

6.  The  radius  c'  df  of  rack-tooth  faces  is  the  same 
length  as  radius  c  d  of  rack-tooth  flanks,  and  is  the 
straight  distance  from  c  to  tooth-thickness  point  b  on 
opposite  side  of  line  O  A. 

7.  The  radius  for  fillet  at  bottom  of  rack  teeth  is 
equal  to  -|-  of  the  widest  part  of  tooth  space.     This 
radius    can   be   varied   to    suit  the  judgment   of  the 
designer,  so  long  as  a  fillet  does  not  interfere  with 
teeth  of  engaging  gear. 


Fig.  14 


Backs  of  the  same  pitch,  to  mesh  with  interchange- 
able gears,  should  be  alike  when  placed  side  by  side, 
and  fit  each  other  when  placed  together  as  in  Fig.  14. 

In  Fig.  13,  a  few  teeth  of  a  15-tooth  wheel  are  shown 
in  mesh  with  the  rack. 


30  BKOWN    &    SHARPE    MFG.    CO. 


CHAPTER  VIII. 

DODBLE-CDRVE  GEARS,  HAVING  MORE  AND  LESS  THAN 
15  TEETH— ANNULAR  GEARS, 


of  Fig.sti5uction  Let  us  draw  two  gears>  ]2  and  24  teeth,  4  P,  in 
mesh.  In  Fig.  15  the  construction  lines  of  the  lower 
or  24-tooth  gear  are  full.  The  upper  or  12-tooth  gear 
construction  lines  are  dotted.  The  line  of  pressure, 
L  P,  and  the  line  K  T  k  answer  for  both  gears.  The 
arcs  A  K  and  O  k  are  described  about  T.  The  radius 
of  these  arcs  is  the  radius  of  pitch  circle  of  a  gear  15 
teeth  4  pitch.  The  length  of  arcs  A  K  and  O  k  is  the 
tooth  thickness  for  4  P.  The  line  K  T  &  is  obtained 
the  same  as  in  Chapter  VII.  for  all  double-curve  gears, 
the  distances  only  varying  as  the  pitch.  Having  drawn 
the  pitch  circles,  the  line  K  T  &,  and,  perpendicular  to 
K  T  &,  the  lines  A  B,  O  r  and  the  line  of  pressure 
L  T  P,  we  proceed  with  the  24-tooth  gear  as  follows : 

1.  From  center  C,  through  r,  draw  line  intersecting 
line  of  pressure  in  m.     Also  draw  line  from  center  C 
to  R,  crossing  the  line  of  pressure  L  P  at  c. 

2.  Through  m  describe  circle  concentric  with  pitch 
circle  about  C.     This  is  the  circle  in  which  to  place 
one  leg  of  dividers  to  describe  flanks  of  teeth. 

3.  The  radius,  m  n,  of  flanks  is  the  straight  distance 
from  m  to  the  first  tooth-thickness  point  on  other  side 
of  line  of  centers,  C  C',  at  v.     The  arc  is  continued  to 
n,  to  show  how  constructed.     This  method  of  obtain- 
ing radius  of  double-curve  tooth  flanks  applies  to  all 
gears  having  more  than  fifteen  teeth. 

4.  The  construction  of  tooth  faces  is  similar  to  15- 
tooth  wheel  in  Chapter  VII.     That  is  :    Draw  a  circle 
through   c    concentric  to   pitch  circle  ;  in  this  circle 
place   one  leg   of   dividers  to    draw  tooth  faces,   the 
radius  of  tooth  faces  being  c  b. 


PROVIDENCE,    E.    I. 


PINION,  12  TEETH, 
GEAR  24  TEETH,  .4  P 

P=4 

N  =12  and  24 

P'=.7854" 

t  =  .3927" 

8  =  .2500" 

D"=  .5000' 

8t/=.2893' 

O--I-/-.5393" 


Cc 

icr.    15. 

DOUBLE  CURVE  GEARS  IN   MESH. 


<"' 'UNIVERSITY 


^S 

:TTT) 


32  BROWN    &    SHARPfi    MFG.    CO. 

of  Kg?ti5l(2onIi      5-  Tbe  radius  of  fillets  at  roots  of  teeth  is  equal  to 
tinued.  one-sixth  the  width  of  space  at  addendum  circle. 

nanks  for  12,     The   constructions  for  flanks  of  12,  13  and  14 
13  and  14  Teeth.'  teeth  are  similar  to  each  other  and  as  follows  : 

1.  Through  center,  C',  draw  line  from  K,  intersecting 
line  of  pressure  in  u.     Through  u  draw  circle  about 
C'.     In  this  circle  one  leg  of  dividers  is  placed  for 
drawing  flanks. 

2.  The  radius  of  flanks  is  the  distance  from  u  to 
the   first  tooth-thickness  point,  e,  on  the  same  side  of 
C  T  C'.     This  gives  convex  flanks.     The  arc  is  con- 
tinued to  V,  to  show  construction. 

3.  This  arc  for  flanks  is  continued  in  or  toward  the 
center,  only  about  one- sixth  of  the  working  depth  (or 
J  s.)  ;  the  lower  part  of  flank  is  similar  to  flanks  of 
gear  in  Chapter  VI. 

4.  The  faces  are  similar  to  those  in  15- tooth  gearr 
Chapter  VII.,  and  to  the  24-tooth  gear  in  the  fore- 
going, the  radius  being  w  y ;  the  arc  is  continued  to  xr 
to  show  construction. 

Annular  Gears.  ANNULAR  GEARS.  Gears  with  teeth  inside  of  a  rim 
or  ring  are  called  Annular  or  Internal  Gears.  The 
construction  of  tooth  outlines  is  similar  to  the  fore- 
going, but  the  spaces  of  a  spur  external  gear  become 
the  teeth  of  an  annular  gear. 

Prof.  MacCord  has  shown  that  in  the  system  just 
described,  the  pinion  meshing  with  an  annular  gear, 
must  differ  from  it  by  at  least  fifteen  teeth.  Thus,, 
a  gear  of  24  teeth  cannot  work  with  an  annular  gear 
of  36  teeth,  but  it  will  work  with  annular  gears  of  39 
teeth  and  more.  An  annular  gear  differing  from  its 
mate  by  less  than  15  teeth  can  be  made.  This  will  be 
shown  in  Part  II. 

Annular-gear  patterns  require  more  clearance  for 
moulding  than  external  or  spur  gears. 

pinions.  In  speaking   of  different- sized  gears,  the   smallest 

ones  are  often  called  "pinions." 

The  angle  of  pressure  in  all  gears  except  involute, 
constantly  changes.  78°  is  the  pressure  angle  in 
double-curve,  or  epicycloidal  gears  for  an  instact 
only ;  in  our  example,  it  is  78°  when  one  side  of  a- 


PROVIDENCE,    E.    I. 


33 


tooth  reaches  the  line  of  centers,  and  the  pressure 
against  teeth  is  applied  in  the  direction  of  the  arrows. 

The  pressure  angle  of  involute  gears  does  not 
change.  An  explanation  of  the  term  angle  of  pressure 
is  given  in  Part  II. 

We  obtain  the  forms  for  epicycloidal  gear  cutters 
by  means  of  a  machine  called  the  Odontom  Engine. 
This  machine  will  cut  original  gears  with  theoretical 
accuracy. 

It  has  been  thought  best  to  make  24  ffear  cutters    24  Double 

curve    Geai 

for  each  pitch,  This  enables  us  to  fill  any  require- 
ment  of  gear-cutting  very  closely,  as  the  range  covered 
by  any  one  cutter  is  so  small  that  it  is  exceedingly  near 
to  the  exact  shape  of  all  gears  so  covered. 

Of  course,  a  cutter  can  be  exactly  right  for  only  one 
gear.  Special  cutters  can  be  made,  if  desired. 


for 


1    PITCH  TOOTH   CURVES 

from  the 
ODONTOM    ENGINE. 


34 


UKOWN    A    SIIAIU'K  WG.   CO. 


CHAPTER   IX. 

BEVEL-&EAR  BLANKS, 


Bevel  Gears  connect  shafts  whose  axes  meet  when 
Bevel °  Gcirsf  suffic^ently  prolonged.  The  teeth  of  bevel  gears  are 
formed  upon  formed  about  the  frustrums  of  cones  whose  apexes 

Irustrums    ol  l 

cones.  arc  at  the  same  point  where  the  shafts  meet.     In  Fig. 

16  we  have  the  axes  A  O  and  B  O,  meeting  at  O,  and 
the  apexes  of  the  cones  also  at  O.  These  cones  are 
called  the  pitch  cones,  because  they  roll  upon  each 
other,  and  because  upon  them  the  teeth  are  pitched. 
If,  in  any  bevel  gear,  the  teeth  were  sufficiently  pro- 
longed toward  the  apex,  they  would  become  infinitely 
small ;  that  is,  the  teeth  would  all  end  in  a  point,  or 
vanish  at  0.  We  can  also  consider  a  bevel  gear  as 
beginning  at  the  apex  and  becoming  larger  and  larger 
as  we  go  away  from  the  apex.  Hence,  as  the  bevel 
gear  keth  are  tapering  from  end  to  end,  we  may  say 


Fly.    16. 


that  a  bevel  gear  has  a  number  of  pitches  and  pitch 
circles,  or  diameters  :  in  speaking  of  the  pitch  of  a 
bevel  gear,  we  mean  always  the  pitch  at  the  largest 


PHOVIDENCE,    11.   I. 


35 


pitch  circle,  or  at  the  largest  pitch  diameter,  as  at 
b  d,  Fig.  17. 

Fig.  17  is  a  section  of  three  bevel  gears,  the  gear 
o  B  q  being  twice  as  large  as  the  two  others.  The 

outer  surface  of  a  tooth  as  m  in'  is  called  the  face  of    Construction 

of  Bevel  Gear 
the  tooth.      The  distance  in  in'  is    usually  called  the  Blanks. 

length  of  the  face  of  the  tooth,  though  the  real  length 
is  the  distance  that  it  occupies  upon  the  line  O  i.  The 
outer  part  of  a  tooth  at  m  n  is  called  its  large  end,  and 
the  inner  part  m'  n'  the  small  end. 

Almost  all  bevel  gears  connect  shafts  that  are  at 
right  angles  with  each  other,  and  unless  stated  other- 
wise we  always  understand  that  they  are  so  wanted. 

The  directions  given  in  connection  with  Fig.  1 7 
apply  to  gears  with  axes  at  right  angles. 

Having  decided  upon  the  pitch 'and  the  numbers  of 
teeth  :— 

1.  Draw  centre  lines  of  shafts,  A  O  B  and  C  O  D, 
at  right  angles. 

2.  Parallel  to  A  O  B,  draw  lines  a  b  and  c  d,  each 
distant  from  A  O  B,   equal  to  half  the   largest  pitch 
diameter  of  one  gear.     For  24  teeth,  4  pitch,  this  half 
largest  pitch  diameter  is  3". 

3.  Parallel  to  COD,  draw  lines  e  f  and  g  h,  dis- 
tant from  COD,  equal   to   half   the   largest   pitch 
diameter  of  the  other  gear.     For  a  gear,  1*2  teeth,  4 
pitch,  this  half  largest  pitch  diameter  is  1|". 

4.  At  the  intersection  of  these  four  lines,  draw 
lines  O  i,  O  j,  O  k,  and  O  1 ;  these  lines  give  the  size 
and  shape  of  pitch  cones.     We  call  them  "•  Cone  Pitch 
Lines." 

5.  Perpendicular  to  the  cone-pitch  lines  and  through 
the  intersection  of  lines  a  b,  c  d,  e  f,   and  g  h,    draw 
lines  m  n,  o  p,  q  r.     We  have  drawn  also  u  v  to  show 
that  another  gear  can  be  drawn  from  the  same  diagram. 
Four  gears,  two  of  each  size,  can  be  drawn  from  this 
diagram. 

6.  Upon  the  lines  m  n,  op,  q  r,  the    addenda  and 
depth  of  the  teeth  are  laid  off,  these    lines  passing 


36 


BROWN   &   SIIARPB  MFG.   CO. 


through  the  largest  pitch  circle  of  the  gears.  Lay  off 
the  addendum,  it  being  in  these  gears  I".  This  gives 
distance  m  n,  o  p,  q  r,  and  u  v  equal  to  the  working 
depth  of  teeth,  which  in  these  gears  is  J".  The 
addendum  of  course  is  measured  perpendicularly  from 
the  cone  pitch  lines  as  at  k  r. 

7.  Draw   lines   O  m,   On,    Op,    O  o,    O  q,    O  r. 
These  lines  give  the  height  of  teeth  above  the  cone- 
pitch   lines   as   they   approach  O,   and  would  vanish 
entirely  at  O.     It  is  quite    as  well  never  to  have  the 
length  of  teeth,  or  face,  m  n'  longer  than  one-third 
the  apex  distance  m  O,  nor  more  than  two  and  one- 
half  times  the  circular  pitch. 

8.  Having  decided  upon  the  length   of  face,  draw 
limiting  lines  m'n'  perpendicular  to  i  O,  q  r  perpen- 
dicular to  k  O,  and  so  on. 

The  distance  between  the  cone-pitch  lines  at  the 
inner  ends  of  the  teeth  m'n'  and  q'  r'  is  called  the  inner 
or  smaller  pitch  diameter,  and  the  circle  at  these  points 
is  called  the  smallest  pitch  circle.  We  now  have  the 
outline  of  a  section  of  the  gears  through  their  axes. 
The  distance  m  r  is  the  whole  diameter  of  the  pinion. 
^nc  distance  q  o  is  the  whole  diameter  of  the  gear. 
Blank's  ca£  be  *n  Pract^ce  these  diameters  can  be  obtained  by  rueasur- 
obtained  by  jnor  the  drawing.  The  diameter  of  pinion  is  3.45"  and 

Measuring 

Drawings.  of  the  gear  G.22".  We  can  find  the  angles  also  by 
measuring  the  drawing  with  a  protractor.  In  the 
absence  of  a  protractor,  teinpletes  can  be  cut  to  the 
drawing.  The  angle  formed  by  line  m  in'  with  a  b  id 
the  angle  of  face  of  pinion,  in  this  pinion  59°  11',  or 
59^°  nearly.  The  lines  q  q'  and  g  h  give  us  angle  of 
face  of  gear,  for  this  gear  22°  19',  or  22j-°  nearly 
The  angle  formed  by  m  navith  a  b  is  called  the  angle 
of  edge  of  pinion,  in  our  sketch  26°  3i',  or  about  26i °. 
The  angle  of  edge  of  gear,  line  q  r  with  g  h,  is  63°  26', 
or  about  63 1°.  In  turning  blanks  to  these  angles  we 
place  one  arm  of  the  protractor  or  templet  against  the 
end  of  the  hub,  when  trying  angles  of  a  blank.  Some 
designers  give  the  angles  from  the  axes  of  gears,  but 


PROVIDENCE,    R.    I. 


37 


38  KKOWX    &   SHARPE   MFG.    CO., 

it  is  not  convenient  to  try  blanks  in  this  way.  The 
method  that  we  have  given  comes  right  also  for  angles 
as  figures  in  compound  rests. 

When  axes  are  at  right  angles,  the  sum  of  angles 
of  edge  in  the  two  gears  equals  90°,  and  the  sums  of 
angle  of  edge  and  face  in  each  gear  are  alike. 

The  angles  of  the  axes  remaining  the  same,  all  pairs 
of  bevel  gears  of  the  same  ratio  have  the  same  angle 
of  edge  ;  all  pairs  of  same  ratio  and  of  same  numbers 
of  teeth  have  the  same  angles  of  both  edges  and  faces 
independent  of  the  pitch.  Thus,  in  all  pairs  of  bevel 
gears  having  one  gear  twice  as  large  as  the  other,  with 
.axes  at  right  angles,  the  angle  of  edge  of  large  gear 
is  63°  26',  and  the  angle  of  edge  of  small  gear  is  26°  34'. 

In  all  pairs  of  bevel  gears  with  axes  at  right  angles, 
one  gear  having  24  teeth  and  the  other  gear  having  12 
teeth,  the  angle  of  face  of  small  gear  is  59°  11'. 
Another      The  following  method  of  obtaining  the  whole  diam- 

method  ot  ob- 
taining Whole  ter  of  bevel  gears  is  sometimes  preferred  : 
Diameter    o  1' 

Blanks.  From  k  lay  off ;  upon  the  cone-pitch  line,  a  distance 

K  w,  equal  to  ten  times  the  working  depth  of  the 
teeth  =  10  D".  Now  add  TV  of  the  shortest  distance 
of  w  from  the  line  g  h,  which  is  the  perpendicular 
dotted  line  w  x,  to  the  outside  pitch  diameter  of  gear, 
and  the  sum  will  be  the  whole  diameter  of  gear.  In 
the  same  manner  TO  of  w  y,  added  to  the  outside  pitch 
diameter  of  pinion,  gives  the  whole  diameter  of  pinion. 
The  part  added  to  the  pitch  diameter  is  called  the 
diameter  increment. 

Part  II  gives  trigonometrical  methods  of  figuring 
bevel  gears :  in  our  Formulas  in  Gearing  there  are 
trigonometrical  formulas  for  bevel  gears,  and  also 
tables  for  angles  and  sizes. 

oMte?eiUGe°r     ^  somewhat  similar  construction  will   do   for  bevel 
Blanks  whose  gears  whose  axes  are  not  at  right  angles. 

Axes    are   not  fe 

"~      *11  *^£*  ^  *ke  axes  are  snown  at   O  B  and  O  D,  the 
angle  BOD  being  less  than  a  right  angle. 

1.  Parallel  to  O  B,  and  at  a  distance  from  it  equal 
to  the  radius  of  the  gear,  we  draw  the  lines  a  b  and  c  d. 


PROVIDENCE,    R.    I. 


39 


40  BKOWN    £   SIIAUPK   MKG.    CO. 

2.  Parallel  to  O  D,  and  at  a  distance  from  it  equal 
to  the  radius  of  the  pinion,  we  draw  the  lines  e  f  and  g  h. 

3.  Now,  through  the  point  j  at  the  intersection  of 
c  d  and  g  h,    we   draw   a   line   perpendicular  to  O  B. 
This  line  k  j,  limited  by  a  b  and  c  d,  represents   the 
largest  pitch  diameter  of  the  gear. 

Through  j  we  draw  a. line  perpendicular  to  O  D. 
This  line  j  1,  limited  by  c  f  and  g  h,  represents  the 
largest  pitch  diameter  of  the  pinion. 

4.  Through  the  point  k  at  the  intersection  of  a  b 
with  k  j,  we  draw  a  line  to  O,  a  line  from  j  to  0,  and 
another  from  1,  at  the  intersection  j  1  and  e  f  to  O. 
These  lines  O  k,  O  j,  and  O  1,  represent  the  cone- 
pitch  lines,  as  in  Fig.  17. 

5.  Perpendicular  to  the  cone-pitch  lines  we  draw 
the  lines  u  v,  o  p,  and  q  r.     Upon  these  lines  we  lay 
off  the  addenda  and  working  depth  as  in  the  previous 
figure,  and  then  draw  lines  to  the  point  O  as  before. 

By  a  similar  construction  Figs.  19  and  20  can  be 
drawn. 


STOCKING    CUTTER. 


MIOVIDENCE,    It.    I.  41 


CHAPTER  X. 

BEVEL   [{EARS- 

FORMS  AND  SIZES  OF  TEETH, 
CUTTING  TEETH, 

To  obtain  the  form  of  the  teeth  in  a  bevel  gear  we      Form    of 

bevel    gear 

do  not  lay  them  out  upon  a  pitch  circle,  as  we  do  in  a  teeth. 
spur  gear,  because  the  rolling  pitch  surface  of  a  bevel 
gear,  at  any  point,  is  of  a  longer  radius  of  curvature 
than  the  actual  radius  of  a  pitch  circle  that  passes 
through  that  point.  Thus  in  Fig.  21,  let  f  g  c  be  a 
cone  about  the  axis  O  A,  the  diameter  of  the  cone 
being  f  c,  and  its  radius  g  c.  Now  the  radius  of 
curvature  of  the  surface,  at  c,  is  evidently  longer  than 
g  c,  as  can  be  seen  in  the  other  view  at  C  ;  the  full 
line  shows  the  curvature  of  the  surface,  and  the  dotted 
line  shows  the  curvature  of  a  circle  of  the  radius  g  c. 
It  is  extremely  difficult  to  represent  the  exact  form  of 
bevel  gear  teeth  upon  a  flat  surface,  because  a  bevel 
gear  is  essentially  spherical  in  its  nature  ;  for  practical 
purposes  we  draw  a  line  c  A  perpendicular  to  O  c, 
letting  c  A  reach  the  centre  line  O  A,  and  take  c  A 
as  the  radius  of  a  circle  upon  which  to  lay  out  the 
teeth.  This  is  shown  at  c  n  m,  Fig.  2*2.  For  con- 
venience the  line  c  A  is  sometimes  called  the  back 
cone  radius. 

Let  us  take,  for  an  example,  a  bevel  gear  and  'a    F*;x",mi>le> 
pinion  -4  and  18  teeth,  5  pitch,  shafts  at  right  angles. 
To  obtain  the  forms  of  the  teeth    and    the   data  for 
cutting,  we  need  to  draw  a  section  of  only  a  half  of 
each  gear,  as  in  Fig.  22. 


liUOWX    *    SHAltl'K    MFG.    CO., 

1.  Draw  the  centre  lines  A  O  and  B  O,  tben  tbe 
lines  g  h  and  c  d,  and  the  gear  blank  lines  as  des- 
cribed in  Chapter  IX.  Extend  the  lines  o'  p'  and  o  p 
until  they  meet  the  centre  lines  at  A'  B'  and  A  B. 

'2.  With  the  radius  A  c  draw  the  arc  c  n  in,  which 
Ave  take  as  the  geometrical  pitch  circle  upon  which  to 
lay  out  the  teeth  at  the  large  end.  The  distance  A'  c' 
is  taken  as  the  radius  of  the  geometrical  pitch  circle 
at  the  small  end  ;  to  avoid  confusion  an  arc  of  this 
circle  is  drawn  at  c  n'  m'  about  A. 

o.  For  the  pinion  we  have  the  radius  B  c  for  the 
geometrical  pitch  circle  at  the  large  end  and  B'  c'  for 
the  small  end :  the  distance  B'  c'  is  transferred  to 
B  c"'. 

4.  Upon  the  arc  cum  lay  off  spaces  equal  to  the 
tooth  thickness  at  the  large  pitch  circle,  which  in  our 
example  is  .3 14".     Draw  the  outlines  of  the  teeth  as 
in  previous  chapters  :  for  single  curve  teeth  we  draw  a 
semi-circle  upon  the  radius  A  c,  and  proceed  as  des- 
cribed in  chapter  III.     For  all  bevel  gears  that  are  to 
be  cut  with  a  rotary  disk  cutter,   or   a   common  gear 
cutter,  single  curve  teeth  are  chosen  ;  and  no  attempt 
should  be  made  to  cut  double  curve    teeth.     Double 
curve  teeth  can   be  drawn  by   the  directions  given  in 
chapters    VII    and    VIII.      We  now  have  the  form  of 
the  teeth  at  the  large  end  of  the   gear.     Repeat  this 
operation  with  the  radius   B  C  about  B,  and  we  have 
the  form  of  the  teeth  at  the  large  end  of  the  pinion. 

5.  The  tooth  parts  at  the  small  end  are  designated 
by  the  same  letters  as  at  the  large,  with  the  addition 
of  an  accent  mark  to  each  letter,  as  in  the  right  hand 
column,  Fig.  2'2,  the  clearance,  f,  however,  is  usually 
the  same  at  the  small  end    as    at   the  large,  for  con- 
venience in  cutting  the  teeth. 

of  the      The  sizes  of  the  tooth  parts  at  the  small  end  are  in 

tooth  |»:irts. 

the  same  proportion  to  those  at  the  large  end  as 
the  line  O  c'  is  to  O  c.  In  our  example  O  c'  is  - ", 
and  O  c  is  3" ;  dividing  O  c'  by  O  c  we  have  -,  or 
.006,  as  the  ratio  of  the  sizes  at  the  small  end  to  those 


PROVIDENCE,  R.  I. 


43 


I 


44  BROWN  &  SIIARPE  MFC!.  CO. 

at  the  large  :  t'  is  .209"  or  f  of  .314",  and  so  on.  If 
the  distance  n  m  is  equal  to  the  outer  tooth  thickness, 
t,  upon  the  arc  cum,  the  lines  n  A  and  m  A  will  be  a 
distance  apart  equal  to  the  inner  tooth  thickness  t' 
upon  the  arc  c"  n  m'.  The  addendum,  s',  and  the 
working  depth,  D'",  are  at  o'  c'  and  o'  p'. 

6.  Upon  the  arcs  c"  n'  m  and  c'"  we  draw  the  forms 
of  the  teeth  of  the  gear  and  pinion  at  the  inside. 
Example  of  As  an  example  of  the  cutting  of  bevel  gears  with 
rotary  disk  cutters,  or  common  gear  cutters,  let  us 
take  a  pair  of  8  pitch,  12  and  24  teeth,  shown  in 
Fig.  23. 

Length  of  In  making  the  drawing  it  is  well  to  remember  that 
tooth  lace.  nolm-no-  is  gained  by  having  the  face  F  E  longer  than 
five  times  the  thickness  of  the  teeth  at  the  large 
pitch  circle,  and  that  even  this  is  too  long  when  it  is 
more  than  a  third  of  the  apex  distance  Q  c.  To  cut  a 
bevel  gear  with  a  rotary  cutter,  as  in  Fig.  24,  is  at 
best  but  a  compromise,  because  the  teeth  change  pitch 
from  end  to  end,  so  that  the  cutter,  being  of  the  right 
form  for  the  large  ends  of  the  teeth  can  not  be  right 
for  the  small  ends,  and  the  variation  is  too  great  when 
the  length  of  face  is  greater  than  a  third  of  the  apex 
distance  O  c,  Fig.  23.  In  the  example,  one-third  of 
the  apex  distance  is  T9F",  but  F  E  is  drawn  only  a 
half  inch,  which  even  though  rather  short,  has  changed 
the  pitch  from  8  at  the  outside  to  finer  than  1 1  nt  the 
inside.  Frequently  the  teeth  have  to  be  rounded  over 
at  the  small  ends  by  filing ;  the  longer  the  teeth  the 
more  we  have  to  file.  If  there  is  any  doubt  about  the 
strength  of  the  teeth,  it  is  better  to  lengthen  at  the 
large  end,  and  make  the  pitch  coarser  rather  than  to 
lengthen  at  the  small  end. 
Data  for  These  data  are  needed  before  beginning  to  cut: 

1 .  The  pitch  and  the  numbers  of  the  teeth  the  same 
as  for  spur  gears. 

2.  The  data  for  the  cutter,  as  to  its  form:  some- 
times two  cutters  are  needed  for  a  pair  of  bevel  gears. 

3.  The  whole  depth  of  the  tooth   spaces,   both  at 


PROVIDENCE,    R.    L 


45 


PINION  18  TEETH. 
GEAR.24.  TEETH.  5  P. 
P  =5. 

N  =1 8  and  24 
F'=    .628" 
t  =    .314" 
8  =     .200" 
D"=     .400" 
S+/=     .231" 
D-+/  =    .431" 


f  =  .209' 
8'=.133' 
D'"=  .266' 
s'+f  =  .165' 
-4-  =.298" 


fiff.  22. 

BEVEL  GEARS,   FORM  AND  SIZE  OF  TEETH, 


46  RKOWN    &    SHAHl'E   MF(i.  CO. 

the  outside  and   inside  ends ;  D"  -j-  f  at  the  outside, 
and  D'"  -f  f  at  the  inside. 

4.  The  thickness  of  the  teeth  at  the  outside  and  at 
the  inside  ;  t  and  t'. 

5.  The  height  of  the  teeth  above  the  pitch  lines  at 
the  outside  and  inside  ;  s  and  s'. 

6.  The  cutting  angles,  or  the  angles  that  the  path 
of  the  cutter  makes  with  the  axes   of  the  gears.     In 
Fig.  23  the  cutting  angle  for  the  gear  c  D  is   A  Op, 
and  the  cutting  angle  for  the  pinion  is  B  O  o. 

selection  of       The  form  of  the  teeth  in  one  of  these  gears  differs 

cutters. 

so  much  from  that  in  the  other  gear  that  two  cutters 
are  required.  In  determining  these  cutters  we  do  not 
have  to  develop  the  forms  of  the  gear  teeth  as  in 
Fig.  22  ;  we  need  merely  measure  the  lines  A  c  and 
B  c,  Fig.  23,  and  calculate  the  cutter  forms  as  if  these 
distances  were  the  radii  of  the  pitch  circles  of  the 
gears  to  be  cut.  Twice  the  length  A  c,  in  inches, 
multiplied  by  the  diametral  pitch,  equals  the  number 
of  teeth  for  which  to  select  a  cutter  tor  the  twenty- 
four- tooth  gear :  this  number  is  about  54,  which  calls 
for  a  number  three  bevel  gear  cutter  in  the  list  of 
bevel  gear  cutters,  page  61.  Twice  B  c,  multiplied 
by  8,  equals  about  13,  which  indicates  a  No.  8  bevel 
gear  cutter  for  the  pinion.  This  method  of  selecting 
cutters  is  based  upon  the  idea  of  shaping  the  teeth  as 
nearly  right  as  practicable  at  the  large  end,  and  then 
filing  the  small  ends  where  the  cutter  has  not  rounded 
them  over  enough. 

In  Fig.  25  the  tooth  L  has  been  cut  to  thickness  at 
both  the  outer  and  inner  pitch  lines,  but  it  must  still 
be  rounded  at  the  inner  end.  The  teeth  M  M  have 
been  filed.  In  thus  rounding  the  teeth  they  should  not 
be  filed  thinner  at  the  pitch  lines. 

There  are  several  things  that  affect  the  shape  of  the 
teeth,  so  that  the  choice  of  cutters  is  not  always  so 
simple  a  matter  as  the  taking  of  the  lines  A  c  and 
B  c  as  radii. 

In  cutting  a  bevel  gear,  in  the  ordinary  gear  cutting 


PKOVIDBNCK.    U.   J. 


47 


48  llllOWN   &   SIIARl'E  MFG.   CO. 

machines,  the  finished  spaces  are  not  always  of  the 
same  form  as  the  cutter  might  be  expected  to  make, 
because  of  the  changes  in  the  positions  of  the  cutter 
and  of  the  gear  blank  in  order  to  cut  the  teeth  of  the 
right  thickness  at  both  ends.  The  cutter  must  of 
course  be  thin  enough  to  pass  through  the  small  end  of 
the  spaces,  so  that  the  large  end  has  to  be  cut  to  the 
right  width  by  adjusting  either  the  cutter  or  the  blank 
sidewise,  then  rotating  the  blank  and  cutting  twice 
around. 

widening       Thus,  in  Fig.  24,  a  gear  and  a  cutter  are  set  to  have 
a  space  widened  at  the  large  end  e',  and  the  last  chip 


to  be  cut  off  by  the  right  side  of  the  cutter,  the  cutter 
having  been  moved  to  the  left,  and  the  blank  rotated 
in  the  direction  of  the  arrow  :  in  a  Universal  Milling 
Machine  the  same  result  would  be  attained  by  moving 
the  blank  to  the  right  and  rotating  it  in  the  direction 
of  the  arrow.  It  may  be  well  to  remember  that  in 
setting  to  finish  the  side  of  a  tooth,  the  tooth  and  the 
cutter  are  first  separated  sidewise,  and  the  blank  is 
then  rotated  by  indexing  the  spindle  to  bring  the  large 
Teeth  nar-  cn(^  °^  tne  tootn  11P  against  the  cutter.  This  tends 
rowed  more  nofc  oniv  to  cut  the  spaces  wider  at  the  large  pitch 

n\   irtC'C  tnun 

at  root.  circle,  but  also  to  cut  off  still  more   at  the  face  of  the 

tooth  ;  that  is,  the  teeth  may  be  cut  rather  thin  at  the 
face  and  left  rather  thick  at  the  root.  This  tendency 
is  greater  as  a  cutting  angle  B  O  o,  Fig.  '23,  is  smaller, 
or  as  a  bevel  gear  approaches  a  spur  gear,  because 
when  the  cutting  angle  is  small  the  blank  must  be 
rotated  through  a  greater  arc  in  order  to  set  to  cut  the 
right  thickness  at  the  outer  pitch  circle.  This  can  be 
understood  by  Figs.  26  and  27.  Fig.  26  is  a  radial- 
toothed  clutch,  which  for  our  present  purpose  can  be 
regarded  as  one  extreme  of  a  bevel  gear  in  which  the 
teeth  are  cut  square  with  the  axis  :  the  dotted  lines 
indicate  the  different  positions  of  the  cutter,  the  side 
of  a  tooth  being  finished  by  the  side  of  the  cutter  that 
is  on  the  centre  line.  In  setting  to  cut  these  teeth 
there  is  the  same  side  adjustment  and  rotation  of  the 


PROVIDENCE,    R.    I. 


Fig.  24 


JI 


50 


BROWN    &    SHARPE   MFG.    CO. 


spindle  as  in  a  bevel  gear,  but  there  is  no  tendency  to 
make  a  tooth  thinner  at  the  face  than  at  the  root.  On 
the  other  hand,  if  we  apply  these  same  adjustments  to 
a  spur  gear  and  cutter,  Fig.  '27,  we  shall  cut  the  face 
F  much  thinner  without  materially  changing  the  thick- 
ness of  the  root  R. 


Fig.  26 

Almost  all  bevel  gears  are  between  the  two  extremes 
of  Figs.  26  and  27,  so  that  when  the  cutting  angle 
B  O  o,  Fig.  23,  is  smaller  than  about  30°,  this  change 
in  the  form  of  the  spaces  caused  by  the  rotation  of  the 
blank  may  be  so  great  as  to  necessitate  the  substitution 


Fly.  28 


PROVIDENCE,    R.    I.  51 

of  a  cutter  that  is  narrower  at  e  e',  Fig.  24,  than  is 
called  for  by  the  way  of  figuring  that  we  have  just 
given :  thus  in  our  own  gear  cutting  department  we 
might  cut  the  pinion  with  a  No.  6  cutter,  instead  of  a 
No.  8.  The  No.  6,  being  for  17  to  20  teeth,  cuts  the 
tooth  sides  with  a  longer  radius  of  curvature  than  the 
No  8,  which  may  necessitate  considerable  filing  at  the 
small  ends  of  the  teeth  in  order  to  round  them  over 
enough.  Fig.  28  shows  the  same  gear  as  Fig.  25,  but 
in  this  case  the  teeth  have  all  been  filed  similar  to 
M  M,  Fig.  25. 

Different  workmen  prefer  different  ways  to  com-  Filing  the 
promise  in  the  cutting  of  a  bevel  gear.  When  a  sman  end1.116 
blank  is  rotated  in  adjusting  to  finish  the  large  end  of 
the  teeth  there  need  not  be  much  filing  of  the  small 
end,  if  the  cutter  is  right,  for  a  pitch  circle  of  the 
radius  B  c,  Fig.  23,  which  for  our  example  is  a  No.  8 
cutter,  but  the  tooth  faces  may  be  rather  thin  at  the 
large  ends.  This  compromise  is  preferred  by  nearly 
.all  workmen,  because  it  does  not  require  much  filing 
of  the  teeth  :  it  is  the  same  as  is  in  our  catalogue  by 
which  we  fill  any  order  for  bevel  gear  cutters,  unless 
otherwise  specified.  This  means  that  we  should  send 
a  No.  8,  8-pitch  bevel  gear  cutter  in  reply  to  an  order 
for  a  cutter  to  cut  the  12-tooth  pinion,  Fig.  23  ;  while 
in  our  own  gear  cutting  department  we  might  cut  the 
same  pinion  with  a  No.  6,  8-pitch  cutter,  because  we 
prefer  to  file  the  teeth  at  the  small  end  after  cutting 
them  to  the  right  thickness  at  the  faces  of  the  large 
end.  We  should  take  a  No.  G  instead  of  a  No.  8  only 
for  a  12-tooth  pinion  that  is  to  run  with  a  gear  two  or 
three  times  as  large.  We  generally  step  off  to  the 
next  cutter  for  pinions  fewer  than  twenty-five  teeth, 
when  the  number  for  the  teeth  has  a  fraction  nearly 
reaching  the  range  of  the  next  cutter :  thus,  if  twice 
the  line  B  c  in  inches,  Fig.  23,  multiplied  by  the 
diametral  pitch,  equals  20.9,  we  should  use  a  No.  5 
cutter,  which  is  for  21  to  25  teeth  inckibive.  In 
filling  an  order  for  a  gear  cutter,  we  do  not  consider 


52  BROWN    &    SHARPE   MFG.    CO. 

the  fraction  but  send  the  cutter  indicated  by  the  whole 
number. 

Later  on  we  will  refer  to-other  compromises  that  are 
made  in  the  cutting  of  bevel  gears. 

The  sizes  of  the  8-pitch  tooth  parts,  Fig.  23,  at  the 
large  end,  are  copied  from  the  table  of  spur  gear 
teeth,  pages  86  to  89. 

The  distance  Oc'  is  seven-tenths   of  the   apex  dis- 
tance Oc,  so  that  the   sizes  of  the  tooth   parts  at  the 
gear  cutting  small  end,  except  f ,    are  seven-tenths  the  large.     The 
order  for  cutting  these  gears  goes  to  the  workmen  in 
this  form  : 

LARGE  GEAR. 

P  =  8 
N  =  24 

D"  +  f '  =  .'210"  D'"  +  f    =  .195" 

t  =  .196"  t'  =  .137" 

s  =  .125"  s'  =  .087" 

Cutting  Angle  =  59°  10' 

SMALL  GEAR. 

N  =  12 

Cutting  Angle  =  22°  18' 

setting  the       ^ig.  32  is  a  side  view  of    a  Gear  Cutting  Machine, 

machine.        ^  beve\  gear  blank  A  is  held  by  the  index  spindle  13. 

The  cutter  C  is  carried   by   the   cutter-slide   D.     The 

cutter-slide-carriage  E  can  be  set  to  the  cutting  angle T 

the  degrees  being  indicated  on  the  quadrant  F. 

Fig.  33  is  a  plan  of  the  machine  :  in  this  view  the 
cutter-slide-carriage,  in  order  to  show  the  details  a 
little  plainer,  is  not  set  to  an  angle. 

Before  beginning  to  cut  the  cutter  is  set  central  with 
the  index  spindle  and  the  dial  G  is  set  to  zero,  so 
that  we  can  adjust  the  cutter  to  any  required  distance 
out  of  centre,  in  either  direction.  Set  the  cutter-slide- 
carriage  E,  Fig.  32,  to  the  cutting  angle  of  the  gear, 
which  for  24-teeth  is  59°  10'  ;  the  quadrant  being 
divided  to  half -degrees,  we  estimate  that  10'  or  J  de~ 


PROVIDENCE,  R.  I. 

gree  more  than  59°.  Mark  the  depth  of  the  cut  at  the 
outside,  as  in  Fig.  30  :  it  is  also  well  enough  to  mark 
the  depth  at  the  inside  as  a  check.  The  thickness  of 
the  teeth  at  the  large  end  is  conveniently  deter- 
mined by  the  solid  gauge,  Fig.  29.  The  gear-tooth 


53 


mg.31 


vernier  caliper,  Fig.  31,  will  measure  the  thickness  of 
teeth  up  to  2  diametral  pitch.  In  the  absence  of  the 
vernier  caliper  we  can  file  a  gauge,  similar  to  Fig  29, 
to  the  thickness  of  the  teeth  at  the  small  end. 

The  index  having  been  set  to  divide  to  the  right  8i£0ofl°toothf 
number  we  cut  two  spaces  central  with  the  blank,  being  finished 
leaving  a  tooth  between  that  is  a  little  too  thick,  as  in 
the  upper  part  of  Fig.  '25.  If  the  gear  is  of  cast  iron, 
and  the  pitch  is  not  coarser  than  about  5  diametral, 
this  is  as  far  as  we  go  with  the  central  cuts,  and  we 
proceed  to  set  the  cutter  and  the  blank  to  finish  first 
one  side  of  the  teeth  and  then  the  other,  going  around 
only  twice.  The  tooth  has  to  be  cut  away  more  in 
proportion  from  the  large  than  from  the  small  end, 
which  is  the  reason  for  setting  the  cutter  out  of  centre, 
as  in  Fig.  24. 


€         OF  THE 
IVERSITY, 
! 


BROWN  &    SI1ARPE   MFGK    CO. 


Fig.  32 


PROVIDENCE,    R.    I.  55 

It  is  important  to  remember  that  the  part  of  the 
cutter  that  is  finishing  one  side  of  a  tooth  at  the  pitch 
line  should  be  central  with  the  gear  blank,  in  order  to 
know  at  once  in  which  direction  to  set  the  cutter  out  of 
centre.  We  can  not  readily  tell  how  much  out  of 
centre  to  set  the  cutter  until  we  have  cut  and  tried, 
because  the  same  part  of  a  cutter  does  not  cut  to  the 
pitch  line  at  both  ends  of  a  tooth.  As  a  trial  distance 
out  of  centre  we  can  take  about  one-tenth  to  one- 
eighth  of  the  thickness  of  the  teeth  at  the  large  end. 
The  actual  distance  out  of  centre  for  the  12-tooth 
pinion  is  .021" :  for  the  24-tooth  gear,  .030%  when 
using  cutters  listed  in  our  catalogue. 

After  a  little  practice  a  workman  can  set  his  blank  ^raSs0* 
the  trial  distance  out  of  centre,  and  take  his  first  cuts, 
without  any  central  cuts  at  all ;  but  it  is  safer  to  take 
central  cuts  like  the  upper  ones  in  Fig.  25.  The 
depth  of  cut  is  partly  controlled  by  the  index-spindle 
raising-dial-shaft  H,  Fig.  33,  which  determines  the 
height  of  the  index  spindle,  and  partly  by  the  position 
of  the  cutter  spindle.  We  now  set  the  cutter  out  of 
centre  the  trial  distance  by  means  of  the  cutter-spindle 
dial-shaft,  I,  Fig.  33.  The  trial  distance  can  be  about 
one-tenth  the  thickness  of  the  tooth  at  the  large  end 
in  a  12-tooth  pinion,  and  from  that  to  one-eighth  the 
thickness  in  a  24-tooth  gear  and  larger.  The  principle 
of  trimming  the  teeth  more  at  the  large  end  than  at 
the  small  is  illustrated  in  Fig.  24,  which  is  to  move 
the  cutter  away  from  the  tooth  to  be  trimmed,  and 
then  to  bring  the  tooth  up  against  the  cutter  by 
rotating  the  blank  in  the  direction  of  the  arrow.  Ad'ustments 

The  rotative  adjustment  of  the  index  spindle  is 
accomplished  by  loosening  the  connection  between  the 
index  worm  and  the  index  drive,  and  turning  the  worm  : 
the  connection  is  then  fastened  again.  The  cutter  is 
now  set  the  same  distance  out  of  centre  in  the  other 
direction,  the  index  spindle  is  adjusted  to  ,trim  the 
other  side  of  the  tooth  until  one  end  is  down  nearly 
to  the  right  thickness.  If  now  the  thickness  of  the 


56  BROWN    &    SHARPE   MFG.    CO. 

small  end  is  in  the  same  proportion  to  the  large  end  as 
Oc'  is  to  Oc,  Fig.  23,  we  can  at  once  adjust  to  trim 
the  tooth  to  the  right  thickness.  But  if  we  find  that 
the  large  end  is  still  going  to  be  too  thick  when  the 
small  end  is  right,  the  out  of  centre  must  be  increased. 

It  is  well  to  remember  this  :  too  much  out  of  centre 
leaves  the  small  end  proportionally  too  thick,  and  too 
little  out  of  centre  leaves  the  small  end  too  thin. 

After  the  proper  distance  out  of  centre  has  been 
learned  the  teeth  can  be  finish-cut  by  going  around  out 
of  centre  first  on  one  side  and  then  on  the  other  with- 
out cutting  any  central  spaces  at  all.  The  cutter 
spindle  stops,  J  J,  can  now  be  set  to  control  the  out 
of  centre  of  the  cutter,  without  having  to  adjust  by 
the  dial  G.  If,  however,  a  cast  iron  gear  is  5-pitch 
or  coarser  it  is  usually  well  to  cut  central  spaces  first 
and  then  take  the  two  out-of -centre  cuts,  going  around 
three  times  in  all.  Steel  gears  should  be  cut  three 
times  around. 

Blanks  are  not  always  turned  nearly  enough  alike  to 
"be  cut  without  a  different  setting  for  different  blanks. 
If  the  hubs  vary  in  length  the  position  of  the  cutter 
spindle  has  to  be  varied.  In  thus  varying,  the  same 
depth  of  cut  or  the  exact  D'  -f-  f  may  not  always  be 
reached.  A  slight  difference  in  the  depth  is  not  so 
objectionable  as  the  incorrect  tooth  thickness  that  it 
may  cause.  Hence,  it  is  well,  after  cutting  once 
around  and  finishing  one  side  of  the  teeth,  to  give 
careful  attention  to  the  rotative  adjustment  of  the 
index  spindle  so  as  to  cut  the  right  thickness. 

After  a  gear  is  cut,  and  before  the  teeth  are  filed,  it 
is  not  always  a  very  satisfactory-looking  piece  of  work. 
In  Fig.  25  the  tooth  L  is  as  the  cutter  left  it,  and  is 
ready  to  be  filed  to  the  sh  ipe  of  the  teeth  M  M,  which 
have  been  filed.  Fig.  34  is  the  pair  of  gears  that  we 
have  been  cutting ;  the  teeth  of  the  12-tooth  pinion 
have  been  filed. 


PROVIDENCE,    R.    I. 


57 


BROWN    &    SHAKPE   MFG.    CO. 

^   second  approximation  in  cutting  with  a   rotary 
tion.  cutter  is  to  widen  the  spaces  at  the  large  end  by  swing- 

ing either  the  index  spindle  or  the  cutter-slide-carriage, 
so  as  to  pass  the  cutter  through  on  an  angle  with  the 
blank  side-ways,  called  the  side-angle,  and  not  rotate 
the  blank  at  all  to  widen  the  spaces.  This  side-angle 
method  is  employed  in  our  No.  2  Automatic  Mitre 
Gear  Cutting  Machine  :  it  is  available  in  the  manufac- 
ture of  mitre  gears  in  large  quantities,  because  with 
the  proper  relative  thickness  of  cutter,  the  tooth-- 
thickness comes  right  by  merely  adjusting  for  the 
side-angle  ;  but  for  cutting  a  few  gears  it  is  not  much 
liked  by  workmen,  because,  in  adjusting  for  the  side- 
angle,  the  central  setting  of  the  cutter  is  usually  lost, 
and  has  to  be  found  by  guiding  into  the  central  slot 
already  cut.  If  the  side-angle  mechanism  pivots  about 
a  line  that  passes  very  near  the  small  end  of  the  tootli 
to  be  cut,  the  central  setting  of  the  cutter  may  not  be 
lost.  With  this  method  '  a  gear  must  be  cut  at  least 
twice  around  ;  in  widening  the  spaces  at  the  large  end, 
the  teeth  are  narrowed  practically  the  same  amount  at 
the  root  as  at  the  face,  so  that  this  side-angle  method 
requires  a  wider  cutter  at  e  e',  Fig.  24,  than  the  first, 
or  rotative  method.  The  amount  of  filing  required 
to  correct  the  form  of  the  teeth  at  the  small  end  is 
about  the  same  as  in  the  first  method. 
A  third  ap.  A  third  approximate  method  consists  in  cutting 

proximation.  ,    , 

the  teeth  right  at  the  large  end  by  going  around  at 
least  twice,  and  then  to  trim  the  teeth  at  the  small  end 
and  toward  the  large  with  another  cutter,  going  around 
at  least  four  times  in  all.  This  method  requires  skill 
and  is  necessarily  a  little  slow,  but  it  contains  possi- 
bilities for  considerable  accuracy. 

A  fourth  ap-  A  f ourth  method  is  to  have  a  cutter  fully  as  thick  as 
the  spaces  at  the  small  end,  cut  rather  deeper  than 
the  regular  depth  at  the  large  end,  and  go  only  once 
around.  This  is  a  quick  method  but  more  inaccurate 
than  the  three  preceding :  it  is  available  in  the  manu- 
facture of  large  numbers  of  gears  when  the  tooth-face 


PROVIDENCE,    R.    I. 


59 


Fig.  34 


60  BROWN    &    SHARPS   MFG.    CO. 

is  shwt  compared  with  the  apex  distance.  It  is  little 
liked,  and  seldom  employed  in  cutting  a  few  gears  :  it 
may  require  some  experimenting  to  determine  the  form 
of  cutter.  Sometimes  the  teeth  are  not  cut  to  the 
regular  depth  at  the  small  end  in  order  to  have  them 
thick  enough,  which  may  necessitate  reducing  the 
addendum  of  the  teeth,  s',  at  the  small  end  by  turning 
the  blank  down.  This  method  is  extensively  employed 
by  chuck  manufacturers. 

A  machine  that  cuts  bevel  gears  with  a  reciprocating 
motion  and  using  a  tool  similar  to  a  planer  tool  is 
called  a  Gear  Planer  and  the  gears  so  cut  are  said  to 
be  planed. 

piamng  of      ^ne  f°rm  of  Gear  Planer  is  that  in  which  the  prin- 

fcevei  gears.    c^^e  emDOcliecl  is  theoretically  correct ;  this  machine 

originates  the  tooth  curves  without  a  former.    Another 

form  of  the  same  class  of  machines  is  that  in  which  the 

tool  is  guided  by  a  former. 

Usually  the  time  consumed  in  planing  a  bevel  gear 
is  greater  than  the  time  necessary  to  cut  the  same  gear 
with  a  rotary  cutter,  thus  proportionately  increasing 
the  cost. 

Pitches  coarser  than  4  are  more  correct  and  some- 
times less  expensive  when  planed  ;  it  is  hardly  prac- 
ticable, and  certainly  not  economical,  to  cut  a  bevel 
gear  as  coarse  as  3P.  with  a  rotary  cutter.  In  gears  as 
fine  as  16P.  planing  affords  no  practical  gain  in  quality. 

While  planing  is  theoretically  correct,  yet  the  wear- 
ing of  the  tool  may  cause  more  variation  in  the  thick- 
ness of  the  teeth  than  the  wearing  of  a  rotary  cutter, 
and  even  a  planed  gear  is  sometimes  improved  by  filing. 
Mounting  of  ^  gears  are  not  correctly  mounted  in  the  place  where 
they  are  to  run,  they  might  as  well  not  be  planed.  In 
fact,  after  taking  pains  in  the  cutting  of  any  gear, 
when  we  come  to  the  mounting  of  it  we  should  keep 
right  on  taking  pains. 

Angles  and      The  method  of  obtaining  the  sizes  and  angles  per- 

Jears.°f  bevel  taining  to  bevel  gears  by  measuring  a  drawing  is  quite 

convenient,    and   with   care   is   fairly   accurate.      Its 


PROVIDENCE,    K.    I. 


accuracy  depends,  of  course,  upon  the  careful  measur- 
ing of  a  good  drawing.  We  may  say,  in  general,  that 
in  measuring  a  diagram,  while  we  can  hardly  obtain 
data  mathematically  exact,  we  are  not  likely  to  make 
wild  mistakes.  Some  years  ago  we  depended  almost 
entirely  upon  measuring,  but  since  the  publication  of 
this  ''Treatise"  and  our  "  Formulas  in  Gearing  "  we 
calculate  the  data  without  any  measuring  of  a  drawing. 
In  the  "  Formulas  in  Gearing"  there  are  also  tables 
pertaining  to  bevel  gears. 

Several  of  the  cuts  and  some  of  the  matter  in  this 
chapter  are  taken  from  an  article  by  O.  J.  Beale,  in 
the  "American  Machinist,"  June  20,  1895. 


1  A* 


CUTTERS  FOR  MITRE  AND  BEVEL 
GEARS. 


Diametral 
Pitch. 

Diameter  of 
Cutter. 

Hole  in 
Cutter. 

4 

3  3-8" 

1  1-4" 

5 

3  1-16 

1  1 

6 

2  3-4 

1  1  16 

8 

2  1-2 

< 

10 

2  1-8 

7  8 

12 

2 

14 

2 

16 

1  15-16 

20 

1  7-8 

24 

1  3-4 

BROWN    &    SHARPE    MFG.    CO. 


CHAPTER  XI. 
WORM  WHEELS— SIZING  BLANKS  OF  32  TEETH  AND  OVER, 


Worm.  ^  WORM  is  a  screw  made  to  mesh  with  the  teeth  of 

a  wheel  called  a  worm-wheel.  As  implied  at  the  end  of 
Chapter  IV.,  a  section  of  a  worm  through  its  axis  is,  in 
outline,  the  same  as  a  rack  of  corresponding  pitch. 
This  outline  can  be  made  either  to  mesh  with  single  or 
double  curve  gear  teeth ;  but  worms  are  usually  made 
for  single  curve,  because,  the  sides  of  involute  rack 
teeth  being  straight  (see  Chapter  IY.),  the  tool  for 
cutting  worm-thread  is  more  easily  made.  The  thread- 
tool  is  not  usually  rounded  for  giving  fillets  at  bottom 
of  worm-thread. 

The  rules  for  circular  pitch  apply  in  the  size  of  tooth 
parts  and  diameter  of  pitch-circle  of  worm-wheel. 

Pitch  of  worm.  The  pitch  of  a  worm  or  screw;  is  usually  given  in  a 
way  different  from  the  pitch  of  a  gear,  viz. :  in  number 
of  threads  to  one  inch  of  the  length  of  the  worm  or 
screw.  Thus,  if  we  say  a  worm  is  2  pitch  we  mean  2 
threads  to  the  inch,  or  the  worm  makes  two  turns  to 
advance  the  thread  one  inch.  But  a  worm  may  be 
double- threaded,  triple- threaded,  and  so  on. 

To  avoid  misunderstanding  it  is  better  always  to 

Worem*Th?elda  cal1  tne  advance  of  the  worm  thread  the  lead.  Thus,  a 
worm-thread  that  advances  one  inch  in  one  turn  we 
call  one -inch  lead  in  one  turn.  A  single- thread  worm 
4  to  V  is  J"  lead.  We  apply  the  term  pitch  to  the  actual 
distance  between  the  threads  or  teeth,  as  in  previous 
chapters.  In  single-thread  worms  the  lead  and  the 
pitch  are  alike.  If  we  have  to  make  a  worm  and  wheel  so 
many  threads  to  one  inch,  we  first  divide  \"  ~by  the  num- 
ber of  threads  to  one  inch,  and  the  quotient  gives  us 
the  circular  pitch.  Hence,  the  wheel  in  Fig.  36  is  J" 

Linear  Pitch,    circular  pitch.    The  term  linear  pitch  expresses   ex- 


PROVIDENCE,    K.    I. 


63 


FIG.  35 -WORM  AND  WORM-WHEEL 

The  thread  of  Worm  is  left-handed ;    Worm  is  single-threaded. 


BROWN   &   SHAEPE   MFG.    CO. 


PROVIDENCE,    R.    I.  65 

actly  what  is  meant  by  circular  pitch.  Linear  pitch 
has  the  advantage  of  being  an  exact  use  of  language 
when  applied  to  worms  and  racks.  The  number  of 
threads  to  one  inch  linear,  is  the  reciprocal  of  the  linear 
pitch. 

Multiply  3.1416  by  the  number  of  threads  to  one 
inch,  and  the  product  will  be  the  diametral  pitch  of  the 
worm-wheel.  Thus,  we  would  say  of  a  double-thread 
worm  advancing  1"  in  1J  turns  that: 

Lead=J"  or  .75".    Linear  pitch  or  P'=-|"  or  .375".    Drawing  of 

Diametral  pitch  or  P=  8.377.    See  table  of  tooth  parts.  Worm-wheel. 

To  make  drawing  of  worm  and  wheel  we  obtain 
data  as  in  circular  pitch. 

1.  Draw  center  line  A  O  and  upon  it  space  off  the 
distance  a  b  equal  to  the  diameter  of  pitch-circle. 

2.  On  each  side  of  these  two  points  lay  off  the  dis- 
tance s,  or  the  usual  addendum=y,  as  b  c  and  b  d. 

3.  From   c  lay  off  the   distance  c  O  equal  to   the 
radius  of  the  worm.     The  diameter  of  a  worm  is  gen- 
erally four  or  five  times  the  circular  pitch. 

4.  Lay  off  the  distances  c  g  and  d  e  each  equal  to  /, 
or  the  usual  clearance  at  bottom  of  tooth  space. 

5.  Through  c  and  e  draw  circles  about  O.     These 
represent  the  whole  diameter  of  worm  and  the  diam- 
eter at  bottom  of  worm-thread. 

6.  Draw  h  O  and  i  O  at  an  angle  of  30°  to  45°  with 
A  O.     These  lines  give  width  of  face  of  worm-wheel. 

7.  Through  g  and  d  draw  arcs  about  O,  ending  in 
h  O  and  i  O. 

This  operation  repeated  at  a  completes  the  outline 
of  worm-wheel.  For  32  teeth  and  more,  the  addendum 
diameter,  or  D,  should  be  taken  at  the  throat  or 
smallest  diameter  of  wheel,  as  in  Fig.  30.  Measure 
sketch  for  whole  diameter  of  wheel-blank. 

The  foregoing  instructions  and  sketch  are  for  cases    Teeth    of 
where  the  teeth  of  the  wheels  are  finished  with  a  hob. 


A  HOB  is  shown   in   Fig.    37,  being  a  steel  piece    Hob. 
threaded  with  the   same  tool  that  threads  the  worm, 
then  grooved  to  make  teeth  for  cutting,  and  hardened. 

The    whole    diameter   of    hob   should   be   at  least  Proportions  of 
2  /,    or  twice  the  clearance  larger  than   the   worm. 


66 


BROWN    &    SHARPE   MFG.    CO. 

In  our  relieved  hobs  the  diameter  is  made  still  larger 
in  order  to  give  the  proper  clearance.     The  outer  cor- 
ners of  hob-thread  can  be  rounded  down  as  far  as  the 
clearance  distance.  The  width  at  top  of  the  hob-thread 
before  rounding  should  be  .31  of  the  linear,  or  circular 
pitch=.31P'.      The  whole  depth  of  thread  should  ba' 
the    ordinary  working   depth    plus   the   clearance= 
D"-f/-  The  diameter  at  bottom  of  hob-thread  should  be 
2/ larger  than  the  diameter  at  bottom  of  worm-thread. 
Por   thread-tool  and  worm-thread  see  end  of  Chapter  IV 
The  thickness  of  cutter  for  grooving  small  hobs,  say  less  than 
two  inches  diameter,  can  be  about  £  the  width  of  thread  at 
top  plus  i"=.£f  -£E'  4.  £".     The  width  of  lands  at  the  bottom 


FIG.  37.— HOB. 

can  be  about  the  depth  of  thread  plus  r=JT+2/+i".  The 
grooves  are  usually  cut  with  a  round  edge  cutter,  the  parallel 
part  of  cutter  just  reaching  the  bottom  of  thread,  making 
the  half-round  bottom  of  grooves  below  the  bottom  of 
thread.  In  small  hobs,  the  teeth  are  often  not  relieved 
between  the  grooves.  In  large  hobs  or  those  more  than 
three  inches  diameter,  the  teeth  may  be  cut  with  radial 
faces,  cutting  the  space  wider  at  the  outer  part  so  as  to  leave 
the  faces  and  backs  of  teeth  about  parallel,  and  the  teeth 
should  be  relieved.  This  can  be  done  in  our  Universal 
Milling  Machine.  A  common  way  in  hobs  two  to  three 
inches  in  diameter,  is  to  relieve  with  a  file. 


PROVIDENCE    11.    I.  67 

The   teeth  of  the   wheel  are  first  cut  as  nearly  to 
the  finished  form  as  practicable ;  the  hob  and  worm- 
wheel  are  mounted  upon  shafts  and  hob  placed  in  mesh 
as  in  Fig.  35.   .  The  hob  is  now  made  to  rotate,  and  is  th J^b  to  use 
dropped  deeper  into  the  wheel  at  each  revolution  of  the 
wheel  until    teeth  are  finished.     The   hob    generally 
drives    the   worm-wheel  during  this   operation.     The 
Universal  Milling  Machine  is  very  convenient  for  doing    Universal 
this  work,  and  with  it  the   distance  between  axes  of  chine  used  in 
worm   and   wheel   can   be   readily    noted.      We  have 
machines  for  nobbing  wheels,  in  which  the  work  spindle 
is    driven    by    gearing   so    that    the   hob    does    not 
have   to   do   the   work   of   driving   the  wheel.      The    wnyaWneei 
object  of  hobbing  a  wheel  is  to  get  more  bearing  sur- 1S 
face  of  the  teeth  upon  worm -thread.    The  worm-wheels, 
Figs.  35  and  43,  were  hobbed.     By  hobbing  we  pro- 
duce outline  of  teeth  something  like  the  thread  of  a  nut. 

If  we  make  the  diameter  of  a  worm-wheel  blank,  that    Worm-Wheel 

i  i  ,1  <-»rv     j       n       T         i-,  -i       Blanks    with 

is  to  nave  less  than  30   teeth,  by   the  common   rules  Less   than    so 

Teeth 

for  sizing  blanks,  and  finish  the  teeth  with  a  hob,  we 
shall  find  the  flanks  of  teeth  near  the  bottom  to  be  un- 
dercut or  hollowing.  This  is  caused  by  the  interfer-  interference 

°  J  of  Thread  and 

ence  spoken  of  in  Chapter  VI.  Thirty  teeth  was  there  Flank, 
given  as  a  limit,  which  will  be  right  when  teeth  are 
made  to  circle  arcs.  With  pressure  angle  75^°,  and 
rack-teeth  with  usual  addendum,  this  interference  of 
rack-teeth  with  flanks  of  gear-teeth  commences  at  31 
teeth  (31TV  geometrically),  and  interferes  with  nearly 
the  whole  flank  in  wheel  of  12  teeth. 

In  Fig  38  the  blank  for  worm-wheel  of  12  teeth  was  Fig.  38. 
sized  by  the  same  rule  as  given  for  Fig.  36.  The  wheel 
and  worm  are  sectioned  to  show  shape  of  teeth  at  the 
mid-plane  of  wheel.  The  flanks  of  teeth  are  undercut 
by  the  hob.  The  worm-thread  does  not  have  a  good 
bearing  on  flanks  inside  of  A,  the  bearing  being  that  of 
a  corner  against  a  surface. 

In  Fig  39  the  blank  for  wheel  was  sized  so  that  pitch-  Fig- 39- 
circle  comes  midway  between  outermost  part  of  teeth 
innermost  point  obtained  by  worm  thread. 


68 


BROWN    &    SHARPE    MFG.   CO. 


Fig.  38. 


PEOVIDENCE,    E.    I. 


69 


Fig.  39. 


70  BROWN    &    SHAEPE    MFG.    CO. 

This  rule  for  sizing  worm-wheel  blanks  has  been  in 
use  to  some  extent.  The  hob  has  cut  away  flanks  of 
teeth  still  more  than  in  Fig.  38.  The  pitch  circle  in 
Fig.  39  is  the  same  diameter  as  the  pitch-circle  in  Fig. 
38.  The  same  hob  was  used  for  both  wheels.  The 
flanks  in  this  wheel  are  so  much  undercut  as  to  mate- 
rially lessen  the  bearing  surface  of  teeth  and  worm- 
thread. 

interference  In  Chapter  VI.  the  interference  of  teeth  in  high- 
numbered  gears  and  racks  with  flanks  of  12  teeth  was 
remedied  by  rounding  off  the  addenda.  Although  it 
would  be  more  systematic  to  round  off  the  threads  of 
a  worm,  making  them,  like  rack-teeth,  to  mesh  with 
interchangeable  gears,  yet  this  has  not  generally  been 
done,  because  it  is  easier  to  make  a  worm-thread  tool 
with  straight  sides. 

Instead  of  cutting  away  the  addenda  of  worm- 
thread,  we  can  avoid  the  interference  with  flanks  of 
wheels  having  less  than  30  teeth  by  making  wheel 
blanks  larger. 

Pig.  40.  The  flanks  of  wheel  in  Fig.  40  are  not  undercut,  be- 

cause the  diameter  of  wheel  is  so  large  that  there  is 
hardly  any  tooth  inside  the  pitch-circle.  The 
pitch-circle  in  Fig.  40  is  the  same  size  as  pitch- 
circles  in  Figs.  38  and  39.  This  wheel  was  sized 
Diameter  at  by  the  following  rule  :  Multiply  ihepitch  diameter  of 

Throat  to  Avoid    ^  °  r  J 

interference,  the  wheel  by  .937,  and  add  to  the  product  four  times 
the  addendum  (4  s) ;  the  sum  will  be  the  diameter  for 
the  blank  at  the  throat  or  small  part.  To  get  the 
whole  diameter,  make  a  sketch  with  diameter  of  throat 
to  the  foregoing  rule  and  measure  the  sketch. 

It  is  impractical  to  hob  a  wheel  of  12  to  about  16  or 
18  teeth  when  blank  is  sized  by  this  rule,  unless  the 
wheel  is  driven  by  independent  mechanism  and  not  by 
the  hob.  The  diameter  across  the  outermost  parts  of 
teeth,  as  at  A  B,  is  considerably  less  than  the  largest 
diameter  of  wheel  before  it  was  hobbed. 

In  general  it  is  well  to  size  all  blanks,  as  by  page  63 
and  Figs.  36  and  38,  when  the  wheels  are  to  be 
hobbed.  Of  course,  if  the  wheel  is  to  be  hobbed  the 


PROVIDENCE,    R.    I. 


71 


Fig.  4O. 


72  BROWN    &    SHARPE    MFG.    CO. 

cutter  should  be  thin  enough  to  leave  stock  for  finish- 
ing. The  spaces  can  be  cut  the  full  depth,  the  cutter 
being  dropped  in. 

To  get  angle  of  worm-thread,  it  is  best  to  apply  pro- 
tractor directly  to  the  thread,  as  computing  the 
angle  affords  but  little  help.  Set  gear  cutter-head  as 
near  the  angle  as  can  be  seen  from  trial  with 
protractor  upon  thread ;  cut  a  few  teeth ;  try  in  worm. 
Generally  the  cutter-head  has  to  be  changed  before 
the  worm  will  take  the  right  position. 

When  worm-wheels  are  not  hobbed  it  is  better  to 
Blank  Like  a  turn  blanks  like  a  spur-wheel.  Little  is  gained  by 

Spur-Wheel.  J 

having  wheels  curved  to  fit  worm  unless  teeth  are  fin- 
ished with  a  hob.  The  teeth  can  be  cut  in  a  straight 
path  diagonally  across  face  of  blank,  to  fit  angle  of 
worm-thread,  as  in  Figs.  41  and  44. 

wheels  for  For  dividing  wheels  to  gear-cutting  engines  the 
Machines.  '*  blanks  are  turned  like  a  spur-wheel  and  a  cutter  about 
T1^-"  larger  diameter  than  the  worm,  is  dropped  in,  as 
in  Figs.  42  and  45,  and  the  worm-thread  is  slightly 
rounded  at  the  outer  corners.  The  radius  for  rounding 
thread  can  be  J  the  width  of  thread  at  the  top. 

Some  mechanics  prefer  to  make  dividing  wheels  in 
two  parts,  joined  in  a  plane  perpendicular  to  axis,  hob 

,  teeth ;  then  turn  one  part  round  upon  the  other,  match 

teeth  and  fasten  parts  together  in  the  new  position, 
and  hob  again  with  a  view  to  eliminate  errors. 

"With  an  accurate  cutting  engine  we  have  found 
wheels  like  Figs.  42  and  45,  not  hobbed,  every  way 
satisfactory.  Dividing  wheels  of  2  feet  diameter  and 
less  are  generally  made  without  arms,  the  part  between 
hub  and  rim  being  a  solid  web.  As  to  the  different 

Figures  43, 44  wheels,  Figs.  43,  44  and  45,  when  worm  is  in  right 
position  at  the  start,  the  life- time  of  Fig.  43,  under 
heavy  and  continuous  work,  will  be  the  longest. 

Fig.  44  can  be  run  in  mesh  with  a  gear  or  a  rack  as 
well  as  with  a  worm  when  made  within  the  angular  limits 
commonly  required.  Strictly,  neither  two  gears  made  in  this 
way,  nor  a  gear  and  a  rack  would  be  mathematically  exact 
as  they  might  bear  on  the  sides  of  the  gear  or  at  the  ends  of 
the  teeth  only  and  not  in  the  middle.  At  the  start  the  con- 


PROVIDENCE,    R.    I. 


73 


Fig.  41. 

"Worm-wheel  with  teeth  cut  in  a  straight  path  diagonally  across  face. 
Worm  is  double-threaded. 


BROWN    &    SHARPE    MFG.    CO. 


Fig.  42. 

Worm  and  Worm- Wheel,  for  Gear-cutting  Engine. 


PKOVIDENCE,    R.    I. 


75 


fuj.  43. 


Fit/.  44. 


.  45. 


76 


BROWN   &  SHARPE   MFG.    CO. 

tact  of  teeth  in  this  wheel  upon  worm-thread  is  in 
points  only:  yet  such  wheels  have  been  many  years 
successfully  used  in  elevators. 

Fig.  45  is  a  neat-looking  wheel.  In  gear  cutting 
engines  where  the  workman  has  occasion  to  turn  the 
work  spindle  by  hand,  it  is  not  so  rough  to  take  hold 
of  as  Figs.  43  and  44.  The  teeth  are  less  liable  to  in- 
jury than  the  teeth  of  Figs  43  and  44. 

Some  designers  prefer  to  take  off  the  outermost  part 
of  teeth  in  wheels  (Figs.  35  and  43),  as  shown  in  these 
two  figures,  and  not  leave  them  sharp,  as  in  Fig.  19. 

We  do  not  know  that  this  serves  any  purpose  except 
a  matter  of  looks. 

In  ordering  worms  and  worm  wheels  the  centre  dis- 
tances should  be  given. 

If  there  can  be  any  limit  allowed  in  the  centre  distance 
it  should  be  so  stated. 

For  instance,  the  distance  from  the  centre  of  a  worm 
to  the  centre  of  a  worm  wheel  might  be  calculated  at 
6"  but  5  31-32"  or  6  1-32"  might  answer. 

By  stating  all  the  limits  that  can  be  allowed,  there 
may  be  a  saving  in  the  cost  of  work  because  time  need 
not  be  wasted  in  trying  to  make  work  within  narrow 
limits  than  need  be. 


HOBS  WITH  RELIEVED  TEETH. 

We  are  prepared  to  make  hobs  of  any  size  with  the 
teeth  relieved  the  same  as  our  gear  cutters.  The  teeth 
can  be  ground  on  their  faces  without  changing  their 
form.  The  hobs  are  made  with  a  precision  screw  so 
that  the  pitch  of  the  thread  is  accurate  before  hard- 
ening. 


PROVIDENCE,    B. 


CHAPTER  XII. 

SIZING  GEARS  WHEN  THE  DISTANCE  BETWEEN  CENTRES  AND  THE 
RATIOS  OF  SPEEDS  ARE  FIXED— GENERAL  REMARKS— WIDTH 
OF  FACE  OF  SPUR  GEARS— SPEED  OF  GEAR  CUTTERS— TABLE 
OF  TOOTH  PARTS. 


Let  us  suppose  that  we  have  two  shafts  14"  apart, 
center  to  center,  and  wish  to  connect  them  by  gears  so  ta£ce  and  Ratio 
that  they  will  have  speed  ratio  6  to  1.  We  add  the  6  flxed- 
and  1  together,  and  divide  14"  by  the  sum  and  get  2" 
for  a  quotient;  this  2",  multiplied  by  6,  gives  us  the 
radius  of  pitch  circle  of  large  wheel  =  12".  In  the  same 
manner  we  get  2"  as  radius  of  pitch  circle  of  small  wheel. 
Doubling  the  radius  of  each  gear,  we  obtain  24 "and 4" 
as  the  pitch  diameters  of  the  two  wheels.  The  two  num- 
bers that  form  a  ratio  are  called  the  terms  of  the  ratio. 
We  have  now  the  rule  for  obtaining  pitch-circle  diame- 
ter of  two  wheels  of  a  given  ratio  to  connect  shafts  a 
given  distance  apart : 

Divide  the  center  distance  by  the  sum  of  the  terms  of   Rule  f^.DU 
the  ratio;  find  the  product  of  twice  the  quotient  by  each  circles. 
term  separately,  and  the  two  products  will  le  the  pitch 
diameters  of  the  two  wheels. 

It  is  well  to  give  special  attention  to  learning  the 
rules  for  sizing  blanks  and  teeth ;  these  are  much 
oftener  needed  than  the  method  of  forming  tooth  out- 
lines. 

A  blank  1^"  diameter  is  to  have  16  teeth:  what  will 
the  pitch  be  ?  What  will  be  the  diameter  of  the  pitch 
circle  ?  See  Chapter  V. 

A  good  practice  will  be  to  compute  a  table  of  tooth 
parts.  The  work  can  be  compared  with  the  tables 
pages  86-89. 


78  BROWN    &    SHARPE   MFG.    CO. 

In  computing  it  is  well  to  take  TT  to  more  than  four 
places,  7t  to  nine  places  =  3.141592653.  ^  to  nine 
places  =  .318309886. 

There  is  no  such  thing  as  pure  rolling  contact  in 
teeth  of  wheels ;  they  always  rub,  and,  in  time,  will 
wear  themselves  out  of  shape  and  may  become  noisy. 

Bevel  gears,  when  correctly  formed,  run  smoother 
than  spur  gears  of  same  diameter  and  pitch,  because 
the  teeth  continue  in  contact  longer  than  the  teeth  of 
spur  gears.  For  this  reason  annular  gears  run  smoother 
than  either  bevel  or  spur  gears. 

Sometimes  gears  have  to  be  cut  a  little  deeper  than 
designed,  in  order  to  run  easily  on  their  shafts.  If 
any  departure  is  made  in  ratio  of  pitch  diameters  it  is 
better  to  have  the  driving  gear  the  larger,  that  is,  cut 
the  follower  smaller.  For  wheels  coarser  than  eight 
diametral  pitch  (8  P),  it  is  generally  better  to  cut  twice 
around,  when  accurate  work  is  wanted,  also  for  large 
wheels,  as  the  expansion  of  parts  from  heat  often  causes 
inaccurate  work  when  cut  but  once  around.  There  is 
not  so  much  trouble  from  heat  in  plainer  web  gears  as 
in  arm  gears. 

width  of  spur  The  width  of  cast-iron  gear  faces  for  general  pur- 
poses can  be  made  to  the  following  rule : 

Divide  8  by  the  diametral  pitch  and  add  J"  to  the 
quotient;  the  sum  will  be  width  of  face  for  the  pitch 
required. 

Example :  What  width  of  face  for  gear  4  P  ?  Divid- 
ing 8  by  4  and  adding  J"  we  obtain  2J",  for  width  of 
face.  For  change  gears  on  lathes,  where  it  is  desira- 
ble not  to  have  face  very  wide,  the  following  rule  can 
be  used : 

Divide  4  by  the  diametral  pitch  and  add  ^". 

By  the  latter  rule  a  4  P  change  gear  would  have  but 
1J"  face. 

speed  of  Gear  The  speed  of  gear  cutters  is  subject  to  so  many  con- 
ditions that  definite  rules  cannot  be  given.  We  append 
a  table  of  average  speeds.  A  coarse  pitch  cutter  for 
pinion,  12  teeth,  would  usually  be  run  slower  than  a 
cutter  for  a  large  gear  of  same  pitch. 


PROVIDENCE,  B.  I. 
TABLE  OF  AVERAGE  SPEEDS  FOR  GEAR-CUTTERS. 


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In  brass  the  speed  of  gear-cutters  can  be  twice  as  B^a£8e  e  d  * n 
fast  as  in  cast  iron.  Clock-makers  and  those  making  a 
specialty  of  brass  gears  exceed  this  rate  even.  A  12  P 
cutter  has  been  run  1,200  (twelve  hundred)  turns  a 
minute  in  bronze.  A  32  P  cutter  has  been  run  7,000 
(seven  thousand)  turns  a  minute  in  soft  brass. 

In  cutting  5  P  cast-iron  gears,  75  teeth,  a  No.  1,  6  P 
cutter  was  run  136  (one  hundred  and  thirty-six)  turns 
a  minute,  roughing  the  spaces  out  the  full  5  P  depth  ; 
the  teeth  were  then  finished  with  a  5  P  cutter,  running 
208  (two  hundred  and  eight)  turns  a  minute,  feeding 
by  hand.  The  cutter  stood  well,  but,  of  course,  the 
cast  iron  was  quite  soft.  A  4  P  cutter  has  finished 
teeth  at  one  cut,  in  cast-iron  gears,  86  teeth,  running  48 
(forty-eight)  turns  a  minute  and  feeding  Ty  at  one 
turn,  or  3  in.  in  a  minute. 

Hence,  while  it  is  generally  safe  to  run  cutters  as  in 
the  table,  yet  when  many  gears  are  to  be  cut  it  is  well  to 
see  if  cutters  will  stand  a  higher  speed  and  more  feed. 

In  gears  coarser  than  4  P  it  is  more  economical  to 
first  cut  the  full  depth  with  a  stocking  cutter  and  then 
finish  with  a  gear  cutter.  This  stocking  cutter  is  made 


80  BROWN    &    SHARPE    MFG.    CO. 

on  the  principle  of  a  circular  splitting  saw  for  wood. 
The  teeth,  however,  are  not  set ;  but  side  relief  is  ob- 
tained by  making  sides  of  cutter  blank  hollowing.  The 
shape  of  stocking  cutter  can  be  same  as  bottom  of 
spaces  in  a  12-tooth  gear,  and  the  thickness  of  cutter 
can  be  J  of  the  circular  pitch,  see  page  40. 

Keep  cutters  The  matter  of  keeping  cutters  sharp  is  so  important 
that  it  has  sometimes  been  found  best  to  have  the  work- 
man grind  them  at  stated  times,  and  not  wait  until  he 
can  see  that  the  cutters  are  dull.  Thus,  have  him 
grind  every  two  hours  or  after  cutting  a  stated  number 
of  gears.  Cutters  of  the  style  that  can  be  ground 
upon  their  tooth  faces  without  changing  form  are  rap- 
idly destroyed  if  allowed  to  run  after  they  are  dull. 
Cutters  are  oftener  wasted  by  trying  to  cut  with  them 
when  they  are  dull  than  by  too  much  grinding.  Grind 
the  faces  radial  with  a  free  cutting  wheel.  Do  not  let 
the  wheel  become  glazed,  as  this  will  draw  the  temper 
of  the  cutter. 

In  Chapter  VI.  was  given  a  series  of  cutters  for  cut- 
ting gears  having  12  teeth  and  more.  Thus,  it  was 
there  implied  that  any  gear  of  same  pitch,  having  135 
teeth,  136  teeth,  and  so  on  up  to  the  largest  gears,  and, 
also,  a  rack,  could  be  cut  with  one  cutter.  If  this  cut- 
ter is  4  P,  we  would  cut  with  it  all  4  P  gears,  having 
135  teeth  or  more,  and  we  would  also  cut  with  it  a  4  P 
rack.  Now,  instead  of  always  referring  to  a  cutter  by 
the  number  of  teeth  in  gears  it  is  designed  to  cut,  it 
has  been  found  convenient  to  designate  it  by  a  letter 
or  by  a  number.  Thus,  we  call  a  cutter  of  4  P,  made 
to  cut  gears  135  teeth  to  a  rack,  inclusive,  No.  1,  4  P. 
We  have  adopted  numbers  for  designating  involute 
involute  Gear  gear-cutters  as  in  the  following  table : 

No.  1  will  cut  wheels  from  135  teeth  to  a  rack  inclusive. 
«     2         "  "  55       "      134  teeth     " 

"     3         "  "  35       "        54     "         " 

ic    4.         «  «  26       "        34     "         " 

«     5         «  '<  21       "        25     "         " 

"     6         "  "  17       "        20     "         " 

«     7         «  «  14       "        16     "         " 

«     8         "  "  12       "        13     "         " 


PROVIDENCE,    R.    I.  SI 

By  tliis  plan  it  takes  eight  cutters  to  cut  all  gears 
having  twelve  teeth  and  over,  of  any  one  pitch. 

Thus  it  takes  eight  cutters  to  cut  all  involute  4  P 
gears  having  twelve  teeth  and  more.  It  takes  eight 
other  cutters  to  cut  all  involute  gears  of  5  P,  having 
12  teeth  and  more.  A  No.  8,  5  P  cutter  cuts  only  5  P 
gears  having  12  and  13  teeth.  A  Nor  6,  10  P  cutter 
cuts  only  10  P  gears  having  17,  18,  19  and  20  teeth. 
On  each  cutter  is  stamped  the  number  of  teeth  at  the 
limits  of  its  range,  as  well  as  the  number  of  the  cutter. 
The  number  of  the  cutter  relates  only  to  the  number 
of  teeth  in  gears  that  the  cutter  is  made  for. 

In  ordering  cutters  for  involute  spur-gears  two  things 
m usb  be  given: 

1.  Either  the  number  of  teeth  to  be  cut  In,  the  c/ear'    now  loonier 

Involuto  Cut- 

or  the  number  of  the  cutter,  as  given  in  the  foregoing  tern. 
table. 

2.  Either  the  pitch  of  the  gear  or  the  diameter  and 
number  of  teeth  to  be  cut  in  the  gear. 

If  25  teeth  are  to  be  cut  in  a  6  P  involute  gear,  the 
cutter  will  be  No.  5,  6  P,  which  cuts  all  6  P  gears  from 
21  to  25  teeth  inclusive.  If  it  is  desired  to  cut  gears 
from  15  to  25  teeth,  three  cutters  will  be  needed,  No. 
5,  No.  6  and  No.  7  of  the  pitch  required.  If  the  pitch 
is  8  and  gears  15  to  25  teeth  are  to  be  cut,  the  cutters 
should  be  No.  5,  8  P,  No.  6,  8  P,  and  No.  7,  8  P. 

For  each  pitch  of  epicycloidal,  or  double-curve  gears,      Kpicycioidai 
24  cutters  are  made.      In  coarse-pitch  gears,  the  varia-  curve  cutters, 
tioii  in  the  shape  of  spaces  between  gears  of  consecu- 
tive-numbered  teeth  is  greater  than  in  fine-pitch  gears. 
A   set   of  cutters   for  each  pitch,  to  consist  of  so  large  a 
number  as   24,  has  been  established  because  double  curve 
teeth  have  generally  been  preferred  in   coarse-pitch   gears, 
though  the  tendency  of  late. years  is  toward   the   involute 
form. 

Our  double  curve  cutters  have  a  guide  shoulder  on  each 
side  for  the  depth  to  cut.  When  this  shoulder  just  reaches 
the  periphery  of  the  blank  the  depth  is  right.  The  marks 
which  these  shoulders  make  on  the  blank,  should  be  as  nar- 
row as  can  be  seen,  when  the  blanks  are  sized  right. 


UKOWN    iz    SHARPS    MFG.    CO. 

Double-curve  gear-cutters  are  designated  by  letters 
instead  of  by  numbers ;  this  is  to  avoid  confusion  in 
ordering. 

Following  is  the  list  of  epicycloidal  or  double-curve 
gear-cutters :  — 
Cutter  A  cuts  12  teeth.     Cutter  M  cuts  27  to  29  teeth. 

"  B  "  is  "       "  N  <•  so «  M  « 

"     C     "     14      "  "      O     "     34  "  37      " 

u     D     u     15      u  u      p      «     33  u  40     « 

"     E     "     16      "  «      Q      «     43  «  -1 9 

u        Ji       u       17          «  u         R        «       50    <;    59 

"  G  "  18      "  "  S  "     GO  "  74 

"  H  "  19      "  "  T  "     75  '•  99 

«'  I  "  20      "  "  U  "  100  ':  149 

"  J  u  21  to  22  "  V  "  150  "  249   (i 

"  K  "  23  to  ^24  "  W  "  250  «  Back. 

"  L  a  24  to  26  u  X  "  Rack. 

A  cutter  that  cuts  more  than  one  gear  is  made  of 
proper  form  for  the  smallest  gear  in  its  range.  Thus, 
cutter  J  for  21  to  22  teeth  is  correct  for  21  teeth; 
cutter  S  for  60  to  74  teeth  is  correct  for  60  teeth, 
and  so  on. 

Epieyoloidal      *n  orc^el^n&  epicycloids!    gear-cutters  designate  the 

cutters.  letter   of  the    cutter  as    in  the  foregoing  table,  also 

either  give  the  pitch  or  give  data  that  will  enable  u-5 
to  determine  the  pitch,  the  same  as  directed  for  invo- 
lute cutters. 

More  care  is  required  in  making  and  adjusting  epi- 
cycloidal gears  than  in  making  involute  gears. 

^  e'TV?  *Gear      ^n  or<lering  bevel  gear  cutters  three  things  must  bo 

cutters.  given  : 

1.  The  number  of  teeth  in  each  gear. 

2.  Either   the  pitch   of  gears  or  the  largest  pitch 

diameter  of  each  gear;  see  Fig.   17. 

3.  The  length  of  tooth 'face. 

If  the  shafts  are  not  to  run  at  right  angles,  it 
should  be  so  stated,  and  the  angle  given.  Involute 
cutters  only  are  used  for  cutting  bevel  gears.  No  at- 
tempt should  be  made  to  cut  epicyclodial  tooth  bevel  gears 
>vrith  rotary  disk  cutters. 


PROVIDENCE,    R.    I.  83 

In  ordering  worm-wheel  cutters,  three  things  must    How  to  order 

Worm  -gear 

be  given :  Cutters. 

1.  Number  of  teeth  in  the  wheel. 

2.  Pitch  of  the  worm;    see    Chapter  JTZ 

3.  Whole  diameter  of  worm. 

In  any  order  connected  with  gears  or  gear-cutters, 
when  the  word  "  Diameter"  occurs,  we  usually  under- 
stand that  the  pitch  diameter  is  meant.  When  the 
ichole  diameter  of  a  gear  is  meant  it  should  be  plainly 
written.  Care  in  giving  an  order  often  saves  the  delay 
of  asking  further  instructions.  An.  order  for  one  gear- 
cutter  to  cut  from  25  to  30  teeth  cannot  be  filled,  be- 
cause iu  takes  two  cutters  of  involute  form  to  cut  from 
25  to  30  teeth,  and  three  cutters  of  epicycloidal  form 
to  cut  from  25  to  30  teeth. 

Sheet  zinc  is  convenient  to  sketch  gears  upon,  and 
also  for  making  templets.  Before  making  sketch,  it  is 
well  to  give  the  zinc  a  dark  coating  with  the  following 
mixture:  Dissolve  1  ounce  of  sulphate  of  copper  (blue 
vitriol)  in  about  4  ounces  of  water,  and  add  about  one- 
half  teaspoonful  of  nitric  acid.  Apply  a  thin  coating 
with  a  piece  of  waste. 

This  mixture  will  give  a  thin  coating  of  copper  to 
iron  or  steel,  but  the  work  should  then  be  rubbed  dry. 
Care  should  be  taken  not  to  leave  the  mixture  where  it 
is  not  wanted,  as  it  rusts  iron  and  steel. 

We  have  sometimes  been  asked  why  gears  are  noisy. 
Not  many  questions  can  be  asked  us  to  which  we  can 
give  a  less  definite  answer  than  to  the  question  why 
gears  are  noisy. 

We  can  indicate  only  some  of  the  causes  which  may 
make  gears  noisy,  such  as  : — depth  of  cutting  not 
right — in  this  particular  gears  are  oftener  cut  too  deep 
than  not  deep  enough  ;  cutting  not  central — this  may 
make  gears  noisy  in  one  direction  when  they  are  quiet 
while  running  in  the  other  direction  ;  centre  distance 
not  right — if  too  deep  the  outer  corners  of  the 
teeth  in  one  gear  may  strike  the  fillets  of  the  teeth 
in  the  other  gear ;  shafts  not  parallel ;  frame  of  the 


BKOWN  &  SIIAKI'K    MF(J.  CO. 

machine  of  such  a  form  as  to  give  oft'  sound  vibrations. 
Even  when  we  examine  a  pair  of  gears  we  cannot 
always  tell  what  is  the  matter. 

NOTE. — For  any  pitch  not  in  the  following  tables  to 
find  corresponding  part : — multiply  the  tabular  value 
for  one  inch  by  the  circular  pitch  required,  and  the 
product  will  be.the  value  for  the  pitch  given.  Exam- 
ple :  What  is  the  value  of  s  for  4  inch  circular  pitch  ? 
.3183  =  s  for  1"  P'  and  .3183  X  4  =  1.2732=s  for  4" 
P. 

NOTE. — For  an  explanation  of  the  expression  j,",  see 
page  17. 

The  expression  ''Addendum  and  1-P'"  (addendum 
and  a  diameter  pitch)  means  the  distance  of  a  tooth 
outside  of  pitch  line  and  also  the  distance  occupied  for 
every  tooth  upon  the  diameter  of  pitch  circle. 


WORM    THREAD    TOOL    GAUGE. 


.269 


8  PITCH 


BROWN?.  SHARPE  MFG.CO, 
PROVIDENCE.  R  X 


DEPTH    OF    GEAR    TOOTH    GAUGE. 


PROVIDENCE,    R.     I, 


85 


GEAR  CUTTERS. 


BROWN    &    SI1ARPE    MFG.    CO. 


GEAE   WHEELS. 

TABLE    OF    TOOTH    PARTS CIRCULAR    PITCH    IN  'FIRST    COLUMN. 


Circular 
Pitch. 

si  . 

^5  "S  oi 

^  s 

Hg 

Diametral 
Pitch. 

Thickness  of 
Tooth  on 
Pitch  Line. 

Addendum 
and  ^ 

"P.  . 

13  »M 

£  . 

9 

III 

P 

i| 

X 

c 

4J 

^  08 

P 

^' 

P 

t 

# 

D" 

*-*-/ 

D"+/ 

P'x.31  P'x.335 

2 

1 
A 

1.5708 
1.6755 

1.0000 
.9375 

.63661.2732  .7366^1.3732 
.5968  L.1937  .69061.2874 

.6200  .6700 
.5813  .6281 

If 
14 

T'T 

1.7952 
1.9333 

.8750 
.8125 

.55701.1141 
.51731  1.0345 

.6445 
.5985 

1.2016 
1.1158 

.5425.5863 
.5038  .5444 

11 

1 

2.0944 

.7500 

.4775  .9549.5525 

1.0299 

.4650  .5025 

IA 

M 

2.1855 

.7187 

.4576 

.9151 

.5294 

.9870 

.4456 

.4816 

is 

T8r 

2.2848 

.6875 

.4377 

.8754  .5064 

.9441 

.4262  .4606 

!A 

it 

2.3936 

.6562 

.4178 

.8:356  .4834 

.9012  .4069 

.4397 

li 

i 

2.5133 

.6250 

.3979 

.7958 

.4604 

.8583  .3875 

.4188 

ifV 

l» 

2.6456 

.5937 

.3780 

.7560 

.4374 

.8156  .3681  .3978 

H 

* 

2.7925 

.5625 

.3581 

.7162 

.4143 

.7724 

.3488  3769 

!A- 

If 

2.9568 

.  5312 

.3382 

.  G7G4 

.3913 

.7295 

.3294 

.  3559 

l 

i 

3.1416 

.5000 

.3183 

.6366 

.3683 

.6866 

.3100 

.3350 

if 

IA 

3.3510 

.4G87 

.2984 

.  5968 

.3453 

.6437 

.2906 

.3141 

1 

H 

3.5904 

.4375'.  2785  .5570J.3223 

.6007 

.2713 

.2931 

•11 

!-& 

3.8G66 

.4062.2586 

.5173  .2993 

.5579 

.2519 

.2722 

i 

!J 

4.1888 

.3750  .2387 

.4775 

.2762 

.5150 

.2325 

.2513 

-H 

1/T 

4.5896 

.34371.  2189 

.4377 

.2532 

.4720 

.2131 

.2303 

* 

1* 

4.7124 

.33331.2122 

.4244 

.2455 

.4577 

.2066 

.2233 

PROVIDENCE,    R,    i 


TABLE  OF  TOOTH  PAETS.—  Continued. 

CIRCULAR    PITCH    IN    FIRST    COLUMN. 


K  _; 

ll 

Threads  or 
Teeth  per  inch 
Linear. 

Diametral 
Pitch. 

IH 

0  ~  V 

III 
|J| 

EH 

Addendum 
and  -J^- 

t  . 
ll 

— 

c  c 

«4-.^OH^ 

Whole  Depth 
of  Tooth. 

«4H  0 

||l 

c-1 

ll 

5  « 

P'   j    g 

P 

£ 

, 

D' 

gjrf 

D"+/ 

P'x.31  P'x.335 

i 

I 

1} 

5.0265 

.3125 

.1989 

.3979 

.2301 

.4291 

.1938 

.2094 

9 
1  0 

If 

2 

5.5851 
6  .  2832 

.2812 
.2500 

.1790 
.1592 

.3581 
.3183 

.2071  .3862 
.1842'.3433 

.1744 
.  1550 

.1884 
.1675 

ft 

2f 

7.1808 

.2187 

.1393 

.2785 

.1611  .3003 

.1356 

.1466 

| 

2i 

7.8540 

.2000 

.1273 

.2546 

.1473  .2746 

.1240 

.1340 

§ 

n 

8.3776 

.1875 

.1194 

.2387 

.1381  .2575 

.1163 

.1256 

i 

3 

9.4248 

.1666 

.1061 

.2122 

.1228  .2289 

.1033 

.1117 

5 
IT 

3<r 

10.0531 

.1562 

.0995 

.1989 

.1151  .2146 

.0969 

.1047 

f 

3i 

10.9956 

.1429 

.0909 

.1819 

.1052  .1962 

.0886 

.0957 

i 

4 

12.5664 

.1250 

.0796 

.1591 

.0921 

.1716 

.0775 

.0838 

f 

4i 

14.1372 

.1111 

.0707 

.1415 

.0818 

.1526 

.0689 

.0744 

i 

5 

15.7080 

.1000 

.0637 

.1273 

.0737 

.  1373 

.0620 

.0670 

iV 

5J 

16.7552 

.0937 

.0597 

.1194 

.0690 

.1287 

.0581 

.0628 

i 

6 

18.8496 

.0833 

.0531 

.1061 

.0614 

.1144 

.0517 

.0558 

4 

7 
8 

21.9911 
25.1327 

0714 
.0625 

.0455 
.0398 

0910 
.0796 

.0526  .0981 
.0460  .0858 

.0443 

.0388 

.0479 
.0419 

1 

y 

9 

28.2743 

.0555 

.0354 

.0707 

.0409  .0763 

.0344 

.0372 

•A- 

10 
16 

31.4159 
50.2655 

.0500 
.0312 

.0318 
.0199 

.0637 
.0398 

.0368L0687 
.0230,  .0429 

.0310 
.0194 

.0335 
.0209 

88 


BROWX    &    SHARPE    MFG.    CO. 


GEAR  WHEELS. 

TABLE  OF  TOOTH  PARTS DIAMETRAL    PITCH   IN    FIRST    COLUMN. 


Diametral 
Pitch. 

Circular 
Pitch. 

Thickness 
of  Tooth  on 
Pitch  Line. 

Jk 

<y  ^ 

Working  Depth 
of  Tooth. 

5 

I* 

P 

P'       * 

s       D' 

*+/. 

D"+f. 

1 

6.2832 

3.1416 

2.0000 

4.0000 

2.3142 

4.3142 

i 

4.1888 

2.0944 

1.3333 

2  .  6666 

1.5428 

2.8761 

i 

3.1416 

1  .  5708 

1.0000 

2.0000 

1.1571 

2.1571 

ij 

2.5133 

1  .  2566 

.8000 

1  .  6000 

.9257 

1.7257 

it 

2.0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

if 

1.7952 

.8976 

.5714 

1  1429 

.6612 

1.2326 

2 

1  .  5708 

.7854 

.5000 

1.0000 

.5785 

1  .  0785 

2J 

1.3963 

.6981 

.4444 

.8888 

.5143 

.9587 

2* 

1.2566 

.6283 

.4000 

.8000 

.4628 

.8628 

2f 

1  .  1424 

.5712 

.3636 

.7273 

.4208 

.7844 

3 

1.0472 

.5236 

.  3333 

.6666 

.3857 

.7190 

31 

.8976 

.4488 

.2857 

.5714, 

.3306 

.6163 

4 

.7854 

.3927 

.2500 

.5000 

.2893 

.5393 

5 

.6283 

.3142 

.2000 

.4000 

.2314 

.4314 

6 

.5236 

.2618 

.  1666 

.3333 

.  1928 

.3595 

7 

.4488 

.2244 

.1429 

.2857 

.1653 

.3081 

8 

.3927. 

.  1963 

.1250 

.2500 

.1446 

.2696 

9 

.3491 

.1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

11 

.2856 

.1428 

.0909 

.1818 

.1052 

.1961 

12 

.2618 

.1309 

0833 

.1666 

.0964 

.1798 

13 

.2417 

.1208 

.0769 

.1538 

.0890 

.1659 

14 

.2244 

.1122 

.0714 

.  1429 

.0826 

.1541 

PROVIDENCE,    R.    I. 


TABLE  OF  TOOTH  PARTS—  Continued. 

DIAMETRAL    PITCH    IN    FIRST    COLUMN. 


Diametral 
Pitch. 

1  ~ 

s 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 
ar.d  1" 

r  r1 

P,,- 

P. 

P'. 

t. 

s. 

D". 

s+f. 

D"+/. 

15 

.2094 

.1047 

.0666 

.1333 

.0771 

.1438 

16 

.1963 

.0982 

.0625 

.1250 

.0723 

.1348 

17 

.1848 

.0924 

.05  8 

.1176 

.0681 

.1269 

18 

.1745 

.0873 

.0555 

.1111 

.0643 

.1198 

19 

.1653 

.0827 

.0526 

.1053 

.0609 

.1135 

20 

.1571 

.0785 

.0500 

.1000 

.0579 

.1079 

22 

.1428 

.0714 

.0455 

.0909 

.0526 

.0980 

24 

.1309 

.0654 

.0417 

.0833 

.0482 

.0898 

26 

28 

.1208 
.1122 

.0604 
.0561 

.0385 
.0357 

.0769 
.0714 

.0445 
.0413 

.0829 
.0770 

30 

.1047 

.0524 

.0333 

.0666 

.0386 

.0719 

32 

.0982 

.0491 

.0312 

.0625 

.0362 

.0674 

34 

.0924 

.0462  1  .0294 

.0588 

.0340 

.0634 

36 

.0873 

.0436 

.0278 

.0555 

.0321 

.0599 

38 

.0827 

.0413 

.0263 

.0526 

.0304 

.0568 

40 

.0785 

.0393 

.0250 

.  0500 

.0289 

.0539 

42 

.0748 

.0374 

.0238 

.0476 

.0275 

.0514 

44 

.0714 

.0357 

.  0227 

.0455 

.0263 

.0490 

46 

.0683 

.0341 

.0217 

.0435 

.0252 

.0469 

J8 

.0654 

.0327 

.0208 

.0417 

.0241 

.0449 

50 

.0628 

.0314 

.0200 

.0400 

.0231 

.0431 

56 

.0561 

.0280 

.0178 

.0357 

.0207 

.0385 

GO 

.  0524 

.  0262 

.0166 

.0333 

.0193 

.0360 

PART    II. 


CHAPTER  I. 
TANGENT  OF  ARC  AND  ANGLE, 


*  be 


In  PART  II.  we  shall  show  how  to  calculate  some  egJ 
of  the  functions  of  a  right-angle  triangle  from  a  table 
of  circular  functions,  the  application  of  these  calcula- 
tions in  some  chapters  of  PART  I.  and  in  sizing  blanks 
and  cutting  teeth  of  spiral  gears,  the  selection  of 
cutters  for  spiral  gears,  the  application  of  continued 
fractions  to  some  problems  in  gear  wheels  and  cutting 
odd  screw-threads,  etc.,  etc. 

A  Function  is  a  quantity  that  depends  upon  another 
quantity  for  its  value.      Thus  the  amount  a  workman 
earns  is  a  function  of  the  time  he  has  worked  and  of  fllfe!J™tion  clev 
his  wages  per  hour. 


In  any  riaht-anale  triangle,  O  A  B,  we  shall,  for 
convenience,  call  the  two  lines  that  form  the  right 
angle  O  A  B  the  sides,  instead  of  base  and  perpen- 
dicular. Thus  O  A  B,  being  the  right  angle  we  call 
the  line  O  A  a  side,  and  the  line  A  B  a  side  also. 

When  we  speak  of  the  angle  A  O  B,  we  call  the  line 
O  A  the  side  adjacent.     When  we  are  speaking  of  thesi(le  ailja<'ont" 
angle  ABO  we  call  the  line    A  B  the  side  adjacent. 
The  line  opposite  the  right  angle  is  the  hypothenuse,  Hypotuenuse. 


92 

Tangent. 


BROWN    &    SHARPE    MFG.    CO. 

The  Tangent,  of  an  arc  is  the  line  that  touches  it  at 
one  extremity  and  is  terminated  by  a  line  drawn  from 
the  center  through  the  other  extremity.  The  tangent 
is  always  outside  the  arc  and  is  also  perpendicular  to 
the  radius  which  meets  it  at  the  point  of  tangency. 


fig.  47. 

Thus,  in  Fig.  46,  the  line  A  B  is  the  tangent  of  the  arc 
A  C.  The  point  of  tangency  is  at  A. 

An  angle  at  the  center  of  a  circle  is  measured  by  the 
arc  intercepted  by  the  sides  of  the  angle.  Hence  the 
tangent  A  B  of  the  arc  A  C  is  also  the  tangent  of  the 
angle  A  O  B. 

In  the  tables  of  circular  functions  the  radius  of  the 
arc  is  unity,  or,  in  common  practice,  we  take  it  as  one 
inch.  The  radius  O  A  being  1",  if  we  know  the  length 
of  the  line  or  tangent  A  B  we  can,  by  looking  in  a 
table  of  tangents,  find  the  number  of  degrees  in  the 
angle  A  O  B. 
TO  find  the  Thus,  if  A  B  is  2.25"  long,  we  find  the  angle  A  O  B 

Degrees  in  an 

Angle.  is  66   very  nearly.     That  is,  having  found  that  2.2460 

is  the  nearest  number  to  2.25  in  the  table  of  tangents 
at  the  end  of  this  volume,  we  find  the  corresponding 
degrees  of  the  angle  in  the  column  at  the  left  hand  of 
the  table  and  the  minutes  to  be  added  at  the  top  of 
the  column  containing  the  2.2460. 

The  table  gives  angles  for  every  10',  which  is  suf- 
ficient for  most  purposes. 


PROVIDENCE,    R.    I.  •'•» 

Now,  if  we  have  a  right-angle  triangle  with  an  angle 
the  same  as  O  A  B,  but  with  O  A  two  inches  long,  the 
line  A  B  will  also  be  twice  as  long  as  the  tangent  of 
angle  A  O  B,  as  found  in  a  table  of  tangents. 

Let  us  take  a  triangle  with  the  side  O  A  =  5"  long,  fi^he^D  e  - 
and  the  side  AB  =  8"  long;  what  is  the  number  of  8™^  iu  au 
degrees  in  the  angle  A  O  B  ? 

Dividing  8"  by  5  we  find  what  would  be  the  length 
of  A  B  if  O  A  was  only  1  '  long.  The  quotient  then 
would  be  the  length  of  tangent  when  the  radius  is  1" 
long,  as  in  the  table  of  tangents.  8  divided  by  5  is 
1.6.  The  nearest  tangent  in  the  table  is  1.6003  and 
the  corresponding  angle  is  58°,  which  would  be  the 
angle  A  O  B  when  A  B  is  8"  and  the  radius  O  A  is  5" 
very  nearly.  The  difference  in  the  angles  for  tangents 
1.6003  and  1.6  could  hardly  be  seen  in  practice.  The 
side  opposite  the  required  acute  angle  corresponds  to 
the  tangent  and  the  side  adjacent  corresponds  to  the 
radius.  Hence  the  rule  : 

To  find  the  tangent  of  either  acute  angle  in  a  right-  Tangent?  the 
angle  triangle  :  Divide  the  side  opposite  the  angle  Inj 
the  side  adjacent  the  angle  and  the  quotient  will  le 
the  tangent  of  the  angle.  This  rule  should  ba  com- 
mitted to  memory.  Having  found  the  tangent  of  the 
angle,  the  angle  can  be  taken  from  the  table  of  tan- 
gents. 

The  complement  of  an  angle  is  the  remainder  after     complement 
subtracting  the  angle  from  90°.     Thus  40°  is  the  com- 
plement of  50°. 

The  Cotangent  of  an  angle  is  the  tangent  of  the  cotangent, 
complement  of  the  angle.  Thus,  in  Fig.  47,  the  line 
A  B  is  the  cotangent  of  A  O  E.  In  right-angle  tri- 
angles either  acute  angle  is  the  complement  of  the 
other  acute  angle.  Hence,  if  we  know  one  acute  angle, 
by  subtracting  this  angle  from  90°  we  get  the  other 
acute  angle.  As  the  arc  approaches  90°  the  tangent 
becomes  longer,  and  at  90°  it  is  infinitely  long. 

The  sign  of  infinity  is  oo.     Tangent  90°  =:  oo. 


94  BROWX    &    SHARPE   MFG.    CO. 

AiT°ieayb°Utthe  -^7  a  table  of  tangents,  angles  can  be  laid  out  upon 
~  j  -  sheet  zinc,  etc.  This  is  often  an  advantage,  as  it  is  not 
convenient  to  lay  protractor  flat  down  so  as  to  mark 
angles  up  to  a  sharp  point.  If  we  could  lay  off  the 
length  of  a  line  exactly  we  could  take  tangents  direct 
from  table  and  obtain  angle  at  once.  It,  however,  is 
generally  better  to  multiply  the  tangent  by  5  or  10 
and  make  an  enlarged  triangle.  If,  then,  there  is  a 
slight  error  in  laying  off  length  of  lines  it  will  not 
make  so  much  difference  with  the  angle. 

Let  it  be  required  to  lay  off  an  angle  of  14°  30'.  By 
the  table  we  find  the  tangent  to  be  .25861.  Multiply- 
ing .25861  by  5  we  obtain,  in  the  enlarged  triangle, 
1.29305"  as  the  length  of  side  opposite  the  angle  14° 
30'.  As  we  have  made  the  side  opposite  five  times  as 
large,  we  must  make  the  side  adjacent  five  times  as 
large,  in  order  to  keep  angle  the  same.  Hence,  Fig. 
48,  draw  the  line  A  B  5"  long  ;  perpendicular  to  this 
line  at  A  draw  the  line  A  O  1.293"  long  ;  now  draw  the 
line  O  B,  and  the  angle  ABO  will  be  14°  30'. 

If  special  accuracy  is  required,  the  tangent  can  be 
multiplied  by  10;  the  line  A  O  will  then  be  2.586//long 
and  the  line  A  B  10"  long.  Remembering  that  the 
acute  angles  of  a  right-angle  triangle  are  the  comple- 
ments of  each  other,  we  subtract  14°  30'  from  90'  and 
obtain  75°  30'  as  the  angle  of  A  O  B. 

The  reader  will  remember  these  angles  as  occurring 
in  PART  I.,  Chapter  IV.,  and  obtained  in  a  different 
way.  A  semicircle  upon  the  line  O  B  touching  the 
extremities  O  and  B  will  just  touch  the  right  angle  at 
A,  and  the  line  O  B  is  four  times  as  long  as  O  A. 

Let  it  be  required  to  turn  a  piece  4"  long,  1"  diam- 
eter at  small  end,  with  a  taper  of  10°  one  side  with  the 
other ;  what  will  be  the  diameter  of  the  piece  at  the 
large  end  ? 

A  section,  Fig.  49,  through  the  axis  of  this  piece  is 
same  as  if  we  added  two  right-angle  triangles,  O 
Ta  peri  n^  g  A  B  and  O'  A'  B',  to  a  straight  piece  A'  A  B  B',  1" 
wide  and  4"  long,  the  acute  angles  B  and  B'  being  5°, 
thus  making  the  sides  O  B  and  O'  B'  10°  with  each 
other. 


TKOVIDENCE,    R.    I. 


95 


-1-.293-+ 
Fiy.  48. 


Fiff.  49. 


r)0  BKOWN    &    SIIAliPE    MFG.    CO. 

The  tangent  of  5°  is  .08748,  winch,  multiplied  by 
4",  gives  .34992"  as  the  length  of  each  line,  A  O  and 
A'  O',  to  be  added  to  1"  at  the  large  end.  Taking 
twice  .34992"  and  adding  to  1"  we  obtain  1.69984"  as 
the  diameter  of  large  end. 

This  chapter  must  be  thoroughly  studied  before 
taking  up  the  next  chapters.  If  ouce  the  memory 
becomes  confused  as  to  the  tangent  and  sine  of  an 
angle,  it  will  take  much  longer  to  get  righted  than  it 
will  to  first  carefully  learn  to  recognize  the  tangent 
of  an  angle  at  once. 

If  one  knows  what  the  tangent  is,  he  can  better  tell 
the  functions  that  are  not  tangents. 


PROVIDENCE,    II.    I. 


CHAPTER   II. 

SINE— COSINE  AND  SECANT :     SOME  OF  THEIR  APPLICATIONS  IN 
MACHINE  CONSTRUCTION. 


Sine    of    Arc 
and  Angle. 


The  Sine  of  an  arc  is  the  line  drawn  from  one 
extremity  of  the  arc  to  the  diameter  passing  through 
the  other  extremity,  the  line  being  perpendicular  to 
the  diameter. 

Another  definition  is :  The  sine  of  an  arc  is  the  dis- 
tance of  one  extremity  of  the  arc  from  the  diameter, 
through  the  other  extremity. 

The  sine  of  an  angle  is  the  sine  of  the  arc  that 
measures  the  angle. 

In  Fig.  50 ,  A  C  is  the  sine  of  the  arc  B  C,  and  of 
the  angle  BOG.  It  will  be  seen  that  the  sine  is 
always  inside  of  the  arc,  and  can  never  be  longer  than 
the  radius.  As  the  arc  ap- 
proaches 90°,  the  sine  comes 
nearer  to  the  radius,  and  at  90° 
the  sine  is  equal  to  1,  or  is  the 
radius  itself.  From  the  defini- 
tion of  a  sine,  the  side  A  C, 
opposite  the  angle  A  O  C,  in 
any  right-angle  triangle,  is  the 
sine  of  the  angle  A  O  C,  when 
O  C  is  the  radius  of  the  arc. 
Hence  the  rule :  In  any  right-angle  triangle,  the  side  TO  mid  the 
opposite  either  acute  angle,  divided  by  the  hypothe- 
nuse,  is  equal  to  the  sine  of  the  angle. 

The  quotient  thus  obtained  is  the  length  of  side 
opposite  the  angle  when  the  hypothenuse  or  radius  is 
unity.  The  rule  should  be  carefully  committed  to 
memory. 


98 


BROWN    &    SHARPE    MFG.    CO. 


Arch°rd  °f  an 


a  straight  line  joining  tlio  extremities  of 
an  arc,  and  is  twice  as  long  as  the  sine  of  half  the 
angle  measured  by  the  arc.  Thus,  in  Fig.  51,  the 
chord  F  C  is  twice  as  long  as  the  sine  A  C. 


Fig. 


Let  there  be  four  holes  equidistant  about  a  circle 
3"  in  diameter  —  Fig.  51  ;  what  is  the  shortest  distance 
between  two  holes  ?  This  shortest  distance  is  the 

flnaxth?chord?  cnord  A  B'  which  is  twice  the  sine  of  the  angle  COB. 
The  angle  A  O  B  is  one-quarter  of  the  circle,  and 
C  O  B  is  one-eighth  of  the  circle.  360°,  divided  by 
8=45°,  the  angle  COB.  The  sine  of  *45°  is  .70710, 
which  multiplied  by  the  radius  1.5",  gives  length  C  B  in  the 
circle,  3"  in  diameter,  as  1.06065".  Twice  this  length  is 
the  required  distance  A  B—  2.1213". 

When  a  cylindrical  piece  is  to  be  cut  into  any  num- 
ber of  sides,  the  foregoing  operation  can  be  applied  to 
obtain  the  width  of  one  side.  A  plane  figure  bounded 

Polygon.  by  straight  lines  is  called  a  polygon. 


PROVIDENCE,    R.    I.  [)9 

When  the  outside  diameter  and  the  number  of  sides  of 
a  regular  polygon  are  given,  to  find  the  length  of 

one  of  the  sides  :    Divide  360°  by  twice  the  number  of    To   ft"'1  the 
.  y    length  of  Side. 

sides  •  multiply  the  sine  of  the  quotient  by  the  outer 

diameter,  and  the  product  will  be  the  length  of  one  of 
the  sides. 

Multiplying  by  the  diameter  is  the  same  as  multi- 
plying by  the  radius,  and  that  product  again  by  2. 

The  Cosine  of  an  angle  is  the  sine  of  the  cornple-  cosiuo. 
ment  of  the  angle. 

In  Fig.  50,  C  O  D  is  the  complement  of  the  angle 
A  O  C ;  the  line  C  E  is  the  sine  of  C  O  D,  and  hence 
is  the  cosine  of  B  0  C.  The  line  O  A  is  equal  to  C  E. 
It  is  quite  as  well  to  remember  the  cosine  as  the  part 
of  the  radius,  from  the  center  that  is  cut  off  by  the 
sine.  Thus  the  sine  A  C  of  the  angle  A  O  C  cuts  off 
the  cosine  O  A.  The  line  O  A  may  be  called  the 
cosine  because  it  is  equal  to  the  cosine  C  E. 

In  any  right-angle  triangle,  the  side  adjacent  either 
acute  angle  corresponds  to  the  cosine  when  the 
hypothenuse  is  the  radius  of  the  arc  that  measures 
tho  angle ;  hence:  Divide  the  side  adjacent  the  acute  TO  rm<i  the 
angle  by  the  hypothenusc,  and  the  quotient  will  be  the 
cosine  of  the  angle. 

When  a  cylindrical  piece  is  cut  into  a  polygon  of 
any  number  of  sides,  a  table  of  cosines  can  be  used  to  Length  of 

sides   of   Poly- 
get  the  diameter  across  the  sides.  s°n- 


100 


BROWN    &    SHARPE    MFG.    CO. 


Let  a  cylinder,  2"  diameter,  Fig.  53,  be  cut  six-sided  : 
what  is  the  diameter  across  the  sides  ? 

The  angle  A  O  B,  at  the  center,  occupied  by  one  of 
these  sides,  is  one-sixth  of  the  circle,  =60°.  The 
cosine  of  one- half  this  angle,  oO°,  is  the  line  C  O ; 
twice  this  line  is  the  diameter  across  the  sides.  The 
cosine  of  30°  is  .86602,  which,  multiplied  by  2,  gives 
1.73204"  as  the  diameter  across  the  sides. 

Of  course,  if  the  radius  is  other  than  unity,  the  cosino 
should  be  multiplied  by  the  radius,  and  the  product 
again  by  2,  in  order  to  get  diameter  across  the  sides ; 
or  what  is  the  same  thing,  multiply  the  cosine  by  the 
whole  diameter  or  the  diameter  across  the  corners. 
ameuter  Across  Tlle  rule  for  obtaining  the  diameter  across  sides  of 
sides  of  a  Poly-  regular  polygon,  when  the  diameter  across  corners  is 
given,  will  then  be:  Multiply  the  cosine  of  360° 
divided  l>y  twice  the  number  of  sides,  by  the  diameter 
across  corners,  and  the  product  will  be  the  diameter 
across  sides. 

Look  at  the  right-hand  column  for  degrees  of  the 
cosine,  and  at  bottom  of  page  for  minutes  to  add  to 
the  degrees. 

The  Secant  of  an  arc  is  a  straight  line  drawn  from 
the  center  through  one  end  of  an  arc,  and  terminated 
by  a  tangent  drawn  from  the  other  end  of  the  arc. 

Thus,  in  Fig.  53,  the  line  O  B  is  the  secant  of  the 
angle  COB. 

A  C  B 


Secaut. 


TO  fl:i,i  the  In  any  right-angle  triangle,  divide  the  hijpothenuse 
by  t/ie  side  adjacent  eitlier  acute  angle^  and  the  quo- 
tient will  be  the  secant  of  that  anyle. 


PROVIDENCE,    R.    I. 


101 


That  is,  if  we  divide  the  distance  OB  by  O  C,  in 
the  right-angle  triangle  COB,  the  quotient  will  be 
the  secant  of  the  angle  COB. 

The  secant  cannot  be  less  than  the  radius ;  it  in- 
creases as  the  angle  increases,  and  at  90°  the  secant  is 
infinity =x  . 

A  six-sided  piece  is  to  be  H"  across  the  sides ;  ho 
large  must  a  blank  be  turned  before  cutting  the  sides  ?  Jf^^oiygan 
Dividing  360°  by  twice  the  number  of  sides,  we  have 
30°,  which  is  the  angle  COB.      The  secant  of  30°  is 
1.1547. 

The  radius  of  the  six-sided  piece  is  .75". 

Multiplying  the  secant  1.1547  by  .75",  we  obtain  the 
length  of  radius  of  the  blank  O  B ;  multiplying  again 
by  2,  we  obtain  the  diameter  1.732"  +  . 

Hence,  in  a  regular  polygon,  when  the  diameter 
across  sides  and  the  number  of  sides  are  given,  to  find 
diameter  across  corners :  Multiply  the  secant  of  360° 
divided  by  twice  the  number  of  sides,  l>y  the  diameter 
across  sides,  and  the  product  will  he  the  diameter 
across  corners. 

It  will  be  seen  that  the  side  taken  as  a  divisor  has 
been  in  each  case  the  side  corresponding  to  the  radius 
of  the  arc  that  subtends  the  angle. 

The  versed  sine  of  an  acute  angle  is  the  part* of 
radius  outside  the  sine,  or  it  is  the  radius  minus  the 
cosine.  Thus,  in.  Fig.  50,  the  versed  sine  of  the  arc 
BC  is  AB.  The  versed  sine  is  not  given  in  the  tables 
of  circular  functions  :  when  it  is  wanted  for  any  angle 
less  than  90°  we  subtract  the  cosine  of  that  angle  from 
the  radius  1.  Having  it  for  the  radius  1,  we  can 
multiply  by  the  radius  of  any  other  arc  of  which  we 
may  wish  to  know  the  versed  sine. 

Fig.  54  is  a  sketch  of  a  gear  tooth  of  IP.  In 
measuring  gear  teeth  of  coarse  pitch  it  is  sometimes  a 
convenience  to  know  the  chordal  thickness  of  the 
tooth,  as  at  ATB,  because  it  may  be  enough  shorter 
than  the  regular  tooth-thickness  AHB,  or  t,  to  require 
attention.  It  may  be  also  well  to  know  the  versed 
sine  of  the  angle  B,  or  the  distance  II,  in  order  to  tell 
where  to  measure  the  chordal  thickness. 


102 


BROWN    &    SIIAKPK    M1XJ      CO. 


PROVIDENCE,  R.  I.  103 

On  pages  104  and  105  are  tables  of  data  pertaining 
to  chordal  thickness  of  IP.  teeth.  For  any  other 
diametral  pitch,  divide  the  number  in  the  tablr  by  that 
pitch. 


104 


BROWX    &    SHAEPE    MFG.    CO. 


CIIORDAL    THICKNESS    OF    TEETH    FOR    GEARS   AND  CUTTERS, 
ON    A    BASIS    OF    1    DIAMETRAL    PITCH. 

N  =  Number  of  teeth  in  gears. 

T  =  Chordal  thickness  of  Tooth.          T  =  I)  sin.  ft' 

H  =  Height  of  Arc.  II  =  K  (1— cos.  #'). 

I)  =  Pitch  Diameter. 

R  =  Pitch  Radius. 

£'  =  90°  divided  by  the  number  of  teeth. 

NOTE.— When  the  tooth  of  a  gear  is  measured,  add  the  height  of  arc  to  ($);   and 
when  gear  cutter  is  measured  subtract  the  height  of  arc  from(S  +  f). 

[NVOLUTE. 


Cutter. 

T         II 

Corrected       Corrected 
S+f  forCuttJs  for  Gear. 

No.l 

—  135T 

—  1 

P  1.5707 

.0047 

1.1524 

1.0047 

..     •? 

—  55  T 

-1 

P  1.5706 

.0112 

1.1459 

1.0112 

-   3 

—  35  T 

-1 

3  11.5702.0176 

1.1395 

1.017(5 

"  4 

—  26  T 

-1 

3  1.5698  .0237 

1.1334 

1.0237 

"  5 

-  21  T 

-1 

P  1.5694 

.0294 

1.1277            1.0294 

u  6 

—  17  T 

—  1 

>  1.5686 

.0362 

1.1  201*            1.03(52 

'fc  7 

—  14  T 

—  U 

P 

1.5675 

.0440 

1.1131 

1.0440 

-   8 

—  12  T 

—  IP 

1.5663 

.0514 

1.1057            1.0514 

11  T 

-1] 

P 

1.5654.0559 

1.1011            1.0559 

10  " 

-1 

P 

1.5643.0616 

1.0955            1.0616 

9T 

-1  P 

1.5(528.0684 

1.0887            1.0684 

8T 

-1 

J 

1.5607.0769 

1.0802            1.07(5!) 

PROVIDENCE,    K.    I. 


105 


Cutter. 

T 

11 

Corrected 
S  +  f  for  Cutt, 

Corrected 
S  for  (iear. 

" 

1  A—   12T—  IP 

1.5663 

.0514 

1.1057 

1.0514 

r>  -    lax—  IP 

1.5670 

.0474 

1.1097 

1.0474 

C  _      14'T-r-lP 

1.5675 

.0440 

1.1131 

1.0440 

D-     15T—  .IP 

1.5679 

.0411 

1.1160 

1.0411 

K-      16  TV-IP 

1.5683 

.0385 

1.1186 

1.0385 

F-      ITT—  IP 

1.5686 

.0362 

1.1209 

1.0362 

G_      1ST—  IP 

1.5688 

.0342 

1.1229 

1.0342 

H_      19'1  —  IP 

1.5690 

.0324 

1.1247 

1.0324 

I  -      20T—  IP 

1.5692 

,0308 

1.1268 

1.0308 

.1  -      21ri  —  IP 

1.5694 

.0294 

1.1277 

1.0294 

K  -      23T—  IP 

1.5696 

.0268 

1.1303 

1.0268 

L  -      25T—  IP 

1.5698 

.0247 

1.1324 

1.0247 

M--   27  T  —  IP 

1.5699 

.0228 

1.1343 

1.0228 

X  _  .    JOT  —  IP 

1.5701 

.0208 

1.1363 

1.0208 

0__    J4T  —  IP 

1.5703 

.0181 

1.1390 

1.0181 

P          58  T  —  IP 

1.S703 

.0162 

1.1409 

1.0162 

Q_     43T—  IP 

1.5705 

.0143 

1.1428 

1.0143 

R  _      50T—  IP 

1.5705 

.0123 

1.1448 

1.0123 

S   -   -    (JO  '  ^  —  1  P 

1.5706 

.0102 

1.1469 

1.0102 

T  -      75  ri  —  IP 

1.5707 

.0083 

1.1488 

1.0083 

1    _KM)ri_.  1  P 

1.5707 

.0060 

1.1511 

1.0060 

v__ir>ori—  IP 

1.5707 

.0045 

1.1526 

1.0045 

\\r—  2r>ori—  i  P 

1.0708 

.0025 

1.1546 

1.0025 

SPECIAL. 


No.  Teeth.          T 

H 

Corrected 
S  +  f  for  Cutt. 

. 
Corrected 
S  for  Gear. 

9T  —  1  P     l.r)i;28 
10T  —  1  P     1.5643 
11T  —  1  P     1.5654 

.0684 
.0616 
.0559 

1.0887 

1.0955 
1.1012 

1.0684 
1.0616 

1.0551) 

Oli  BROWN    Si    SIIAIU'K    MFC*.    CO. 

'CHAPTER   III. 

APPLICATION  OF  CIRCULAR  FUNCTIONS— WHOLE  DIAMETER  OF 
BEVEL  GEAR  BLANKS— ANGLES  OF  BEVEL  GEAR  BLANKS. 


The  rules  given  in  this  chapter  apply  only  to  bevel 
gears  having  the  center  angle  c  O  i  not  greater  than  !)0C>. 

To  avoid  confusion  we  will  illustrate  one  gear  only. 
The  same  rules  apply  to  all  sizes  of  bevel  gears.  Fig. 
55  is  the  outline  of  a  pinion  4  P,  20  teeth,  to  mesh  with 
a  gear  28  teeth,  shafts  at  right  angles.  For  making 
sketch  of  bevel  gears  see  Chapter  IX.,  PART!. 

In  Fig.  55,  the  line  O  in'  m  is  continued  to  the  line 
a  I.  The  angle  c'  O  i  that  the  cone  pitch-line  makes 
with  the  center  line  may  be  called  the  center  angle. 
Angle  of  The  center  angle  c'  O  i  is  equal  to  the  angle  of  edge 

Fig.  55-     ,    .  ,    .  .  .  , 

c  i  c.  c  i  is  tne  side  opposite  the  center  angle  c  O 
i,  and  c'  O  is  the  side  adjacent  the  center  angle,  c' 
i  =  2.5";  c'  O  =  3.5".  Dividing  2.5"  by  3.5"  we 
obtain  .71428"  +  as  the  tangent  of  c'  O  i.  In  the  table 
we  find  .71329  to  be  the  nearest  tangent,  the  corre- 
sponding angle  being  35°  30'.  35J-0,  then,  is  the  center 
angle  c  O  i  and  the  angle  of  edge  c  i  n.  very  nearly. 

When  the  axes  of  bevel  gears  are  at  right  angles  the 
angle  of  edge  of  one  gear  is  the  complement  of  angle 
of  edge  of  the  other  gear.  Subtracting,  then,  35^° 
from  90°  we  obtain  54£°  as  the  angle  of  edge  of  gear 
28  teeth,  to  mesh  with  gear  20  teeth.  Fig.  55.  from  which  we 
have  the  rule  for  obtaining  centre  angles  when  the  axes  of 
gears  are  at  right  angles. 

Divide  the  radius  of  the  pinion  by  the  radius  of  the  gear 
and  the  quotient  will  be  the  tangent  of  centre  angle  of  the 
pinion. 

Now  subtract  this  centre  angle  from  90  deg.  and  we  have 
the  centre  angle  of  the  gear. 

The  same  result  is  obtained  by  dividing  the  number  of 
teeth  in  the  pinion  by  the  number  of  teeth  in  the  gear;  the 
quotient  i;-:  the  tangent  of  the  centre  angle. 


PROVIDENCE,    K.    I. 


107 


10S  BROWN    &    SHARPE    MFG.    CO, 

Angle  of  Face.  TO  obtain  angle  of  face  O  m"  c',  the  distance  c'  O 
becomes  the  side  opposite  and  the  distance  m"  c  is 
the  side  adjacent. 

The  distance  c'  O  is  3.5",  the  radius  of  the  28  tooth 
bevel  gear.  The  distance  c  m"  is  by  measurement 
2.82". 

Dividing  3.5  by  2.82  we  obtain  1.2411  for  tangent 
of  angle  of  face  O  m"  c'.  The  nearest  tangent  in  the 
table  is  1.2422  and  the  corresponding  angle  is  51°  10'. 
To  obtain  cutting  angle  c  O  n"  we  divide  the  distance 
c'  n"  by  c  O.  By  measurement  c'  n'  is  2.2".  Divid- 
ing 2.2  by  3.5  we  obtain  .62857  for  tangent  of  cutting 
angle.  The  nearest  corresponding  angle  in  the  table 
is  32°10'. 

The  largest  pitch  diameter,  kj,  of  a  bevel  gear,  as  in 
Fig.  56,  is  known  the  same  as  the  pitch  diameter  of 
any  spur  gear.  Now,  if  we  know  the  distance  1>  o  or 
its  equal  a  q,  we  can  obtain  the  whole  diameter  of 
bevel  gear  blank  by  adding  twice  the  distance  I  o  lo 
the  largest  pitch  diameter. 

Twice  the  distance  b  o,  or  what  is  the  same  thing, 
the  sum  of  a  q  and  bo  is  called  the  diameter  incre- 
ment, because  it  is  the  amount  by  which  we  increase 
the  largest  pitch  diameter  to  obtain  the  whole  or  out- 
side diameter  of  bevel  gear  blanks.  The  distance  b  o 
can  be  calculated  without  measuring  the  diagram. 

The  angle  b  o  j  is  equal  to  the  angle  of  edge. 

The  angle  of  edge,  it  will  be  remembered,  is  the 
angle  formed  by  outer  edge  of  blank  or  ends  of  teeth 
with  the  end  of  hub  or  a  plane  perpendicular  to  the 
axis  of  gear. 

The  distance  b  o  is  equal  to  the  cosine  of  angle  of 
edge,  multiplied  by  the  distance^'  o.  The  distance  j  o 
is  the  addendum,  as  in  previous  chapters  ( =  s). 

Hence  the  rule  for  obtaining  the  diameter  increment 
of  any  bevel  gear:  Multiply  the  cosine  of  angle  of 
edge  by  the  working  depth  of  teeth  (D*),  and  the 
product  will  be  the  diameter  increment. 

By  the  method  given  in  Chapter  II.  we  find  the 
angle  of  edge  of  gear  (Fig.  56)  is  56°  20'.  The  cosine 
of  56°  20°  is  .55430,  which,  multiplied  by  |",  or  the 
-  depth  of  the  3  P  gear,  gives  the  diameter  increment  of 
the  bevel  gear  18  teeth,  3  P  meshing  with  pinion  of  12 


:?KOYIDENCE,    11.    I. 


109 


)  1 0  BROWN    &    SHARPE    MFG.    CO. 

teeth.  I  of  .55436^.369"  -f  (or  .37",  nearly).  Adding 
the  diameter  increment,  .37",  to  the  largest  pitch 
diameter  of  gear,  6",  we  have  6.37"  as  the  outside 
diameter. 

In  the  same  manner,  the  distance  c  d  is  half  the 
diameter  increment  of  the  pinion.  The  angle  c  d  k  is 
equal  to  the  center  angle  of  pinion,  and  when  axes  are 
at  right  angles  is  the  complement  of  center  angle  of 
gear.  The  center  angle  of  pinion  is  33°  40'.  The 
cosine,  multiplied  by  the  working  depth,  gives  .555" 
for  diameter  increment  of  pinion,  and  we  have  4.555" 
for  outside  diameter  of  pinion. 

In  turning  bevel  gear  blanks,  it  is  sufficiently  accu- 
rate to  make  the  diameter  to  the  nearest  hundredth  of 
an  inch. 

Angle  incre  The  small  angle  o  Oj  is  called  the  angle  increment,. 
When  shafts  are  at  right  angles  the  face  angle  of  one 
gear  is  equal  to  the  center  angle  of  the  other  gear, 
minus  the  angle  increment. 

Thus  the  angle  of  face  of  gear  (Fig.  56)  is  less  than 
the  center  angle  D  O  &,  or  its  equal  O  j  k  by  the  angle 
o  O  j.  That  is,  subtracting  o  O  j  from  O  j  k,  the  re- 
mainder will  be  the  angle  of  face  of  gear. 

Subtracting  the  angle  increment  from  the  center 
angle  of  gear,  the  remainder  will  be  the  cutting 
angle. 

The  angle  increment  can  be  obtained  by  dividing 
oj,  the  Bide  opposite,  by  O^',  the  side  adjacent,  thus 
finding  the  tangent  as  usual. 

The  length  of  coiie-pitch  line  from  the  common 
center,  O  to  j,  can  be  found,  without  measuring  dia- 
gram, by  multiplying  the  secant  of  angle  Oj  /v,  or  tho 
center  angle  of  pinion,  by  the  radius  of  largest  pitch 
diameter  of  gear. 

The  secant  of  angle  Oj  k,  33°  40',  is  1.2015,  which, 
multiplied  by  3",  the  radius  of  gear,  gives  3.6045"  its 
the  length  of  line  O  j. 

Dividing  oj  by  Oj,  we  have  for  tangent  .0924,  and 
for  angle  increment  5°  20'. 

The  angle  increment  can  also  be  obtained  by  the 
following  rule : 


PROVIDENCE,    R.    I.  Ill 

Divide  the  sine  of  center  angle  by  half  the  nun* 
ber  of  teeth,  and  the  quotient  will  be  the  tangent  of 
increment  angle. 

Subtracting  the  angle  increment  from  center  angles 
of  gear  and  pinion,  we  have  respectively : 

Cutting  angle  of  gear,  51°. 

Cutting  angle  of  pinion,  28°  20'. 
.Remembering  that  when   the   shafts    are    at   right 
angles,  the  face  angle  of  a  gear  is  equal  to  the  cutting 
angle  of  its  mate  (Chapter  X.  part  1),  we  have: 

Face  angle  of  gear,  28°  20'. 

Face  angle  of  pinion,  51°. 

It  will  bs  seen  that  both  the  whole  diameter  and  the 
angles  of  bevel  gears  can  be  obtained  without  making 
a  diagram.  Mr.  George  B.  Grant  has  made  a  table  of 
different  pairs  of  gears  from  1  to  1  up  to  10  to  1,  con- 
taining diameter  increments,  angle  increments  and 
center  angles,  and  has  published  it  in  the  American 
Machinist  of  October  31, 1885.  We  have  adopted  the 
terms  "  diameter  increment,"  "  angle  increment  "  and 
"center  angle"  from  him.  He  uses  the  term  "back 
angle"  for  what  we  have  called  angle  of  edge,  only  he 
measures  the  angle  from  the  axis  of  the  gear,  instead  Tolayoutan 
of  from  the  side  of  the  gear  or  from  the  end  of  hub,  Angle  by  the 
as  we  have  done ;  that  is,  his  k'back  angle"  is  the  com- 
plement of  our  angle  of  edge. 

In  laying  out  angles,  the  following  method  may  be 
preferred,  as  it  does  away  with  the  necessity  of  making 
a  right  angle :  Draw  a  circle,  ABO  (Fig.  57),  ten 


112 


BROWN   &    SHARPE   MFG.    CO. 

inches  in  diameter.  Set  the  dividers  to  ten  times  the 
sine  of  the  required  angle,  and  point  off  this  distance 
in  the  circumference  as  at  A  B.  From  any  point  O  in 
the  circumference,  draw  the  lines  O  A  and  O  B.  The 
angle  A  O  B  is  the  angle  required.  Thus,  let  the  re- 
quired angle  be  12°.  The  sine  of  12°  is  .20791,  which, 
multiplied  by  10,  gives  2.0791",  or  2-j-J-^"  nearly,  for 
the  distance  A  B. 

Any  diameter  of  circle  can  be  taken  if  we  multiply 
the  sine  by  the  diameter,  but  10"  is  very  convenient, 
as  all  we  have  to  do  with  the  sine  is  to  move  the 
decimal  point  one  place  to  the  right. 

If  either  of  the  lines  pass  through  the  centre,  then  tho 
two  lines  which  do  not  pass  through  the  centre  will  form  a 
right  angle.  Thus,  if  0  B  passes  through  the  centre  then 
the  two  lines  A  B  and  A  0  will  form  a  right  angle  at  A. 


PROVIDENCE,    R.    I.  Ho 


CHAPTER  IV. 
SPIRAL  GEARS— CALCULATIONS  FOR  PITCH  OF  SPIRALS, 


When  the  teeth,  of  a  gear  are  cut,  not  in  a  straight  si)iral  (Joar- 
path,  like  a  spur  gear,  but  in  a  helical  or  screw-like 
path,  the  gear  is  called,  technically,  a  twisted  or  screw 
gear,  but  more  generally  among  mechanics,  a  spiral 
gear.  A  distinction  is  sometimes  made  between  a 
screw  gear  and  a  twisted  gear.  In  twisted  gears  the 
pitch  surfaces  roll  upon  each  other,  exactly  like  spur 
gears,  the  axes  being  parallel,  the  same  as  in  Fig.  1, 
Part  I.  In  screw  gears  there  is  an  end  movement, 
or  slipping  of  the  pitch  surfaces  upon  each  other,  the 
axes  not  being  parallel.  In  screw  gearing  the  action 
is  analogous  to  a  screw  and  nut,  one  gear  driving 
another  by  the  end  movement  of  its  tooth  path.  This 
is  readily  seen  in  the  case  of  a  worm  and  worm-wheel, 
when  the  axes  are  at  right  angles,  as  the  movement  of 
wheel  is  then  wholly  due  to  the  end  movement  of 
worm  thread.  But,  as  we  make  the  axes  of  gears  more 
nearly  parallel,  they  may  still  be  screw  gears,  but  the 
distinction  is  not  so  readily  seen. 

We  can  have  two  gears  that  are  alike  run  together, 
with  their  axes  at  right  angles,  as  at  A  B,  Fig.  59. 

The  same  gear  may  be  used  in  a  train  of  screw  gears 
or  in  a  train  of  twisted  gears.  Thus,  B,  as  it  relates  to 
A,  may  be  called  a  screw  gear ;  but  in  connection  with 
C,  the  same  gear,  B,  may  be  called  a  twisted  gear. 
These  distinctions  are  not  usually  made,  and  we  call 
all  helical  or  screw-like  gears  made  011  the  Universal 
Milling  Machine  spiral  geurs. 

When  two  external  spiral  o-ears  run  together,  with     Direction  of 

,,      .  Spiral  with  ref- 

tneir  axes  parallel,  the  teeth  oi  the  gears  must  have  ereuce  to  Axes. 

.,      ,          ,         .      ,  Fig.  59. 

opposite  hand  spirals. 


BROWN    &    SHARPE   MFG.    CO. 

Thus,  in  Fig.  59  the  gear  B  has  right  hand  spiral 
teeth,  and  the  gear  C  has  left  hand  spiral  teeth.  When 
the  axes  of  two  spiral  gears  are  at  right  angles,  both 
gears  must  have  the  same  hand  spiral  teeth.  A  and 
B,  Fig.  59,  have  right  hand  spiral  teeth.  If  both  gears 
A  and  B  had  left  hand  spiral  teeth,  the  relative  direc- 
tion in  which  they  turn  would  be  reversed, 
spiral  Pitch.  The  spiral  pitch  Ql,  pitch  of  spiral  is  tlie  Distance  the 

spiral  advances  in  one  turn.  Strictly,  this  is  the  lead 
of  the  spiral.  A  cylinder  or  gear  cut  with  spiral 
grooves  is  merely  a  screw  of  coarse  pitch  or  long  lead ; 
that  is,  a  spiral  is  a  coarse  pitch  screw,  and  a  screw  is 
a  fine  pitch  spiral. 

Since  the  introduction  and  extensive  use  of  the 
Universal  Milling  Machine,  it  has  become  customary 
to  call  any  screw  cut  in  the  milling  machine  a  spiral. 
The  spiral  pitch  is  given  as  so  many  inches  to  one 
turn.  Thus,  a  cylinder  having  a  spiral  groove  that  ad- 
vances six  inches  to  one  turn,  is  said  to  have  a  six  inch 
spiral. 

In  screws  the  pitch  is  often  given  as  so  many  threads 
to  one  inch.  Thus,  a  screw  of  •£"  lead  is  said  to  be 
2  threads  to  the  inch.  The  reciprocal  expression  is 
not  much  used  with  spirals.  For  example,  it  would 
not  be  convenient  to  speak  of  a  spiral  of  6"  lead,  as  £ 
threads  to  one  inch. 

The  calculations  for  spirals  are  made  from  the  func- 
tions of  a  right  angle  triangle. 

Example,      Cut   from    paper    a    riirht    anii'le    triangle,    one   side   of 
showing  th  e  na-  . 

ture  of  a  Helix  the    right    angle     (>      long,     and    the    other   side    ot    the 

right  angle  2".  Make  a  cylinder  6"  in  circumference. 
It  will  be  remembered  (Part  I.,  Chapter  II.)  that  the 
circumference  of  a  cylinder,  multiplied  by  .3183,  equals 
the  diameter— 6" x. 3183=1.9098".  Wrap  the  paper 
triangle  around  the  cylinder,  letting  the  2"  side  be 
parallel  to  the  axis,  the  6"  side  perpendicular  to  the 
axis  and  reaching  around  the  cylinder.  The  hypoth- 
eneuse  now  forms  a  helix  or  screw-like  line,  called 
a  spiral.  Fasten  the  paper  triangle  thus  wrapped 
around.  See  Fig.  60, 


PROVIDENCE,    K.    I. 


115 


E 


FIG.  58-RACKS  AND  GEARS, 


!& SH  AFlFE    MF?G.    CO. 


"-'.;,',:":.,'•',.,„.  ,     »• 


FIG,  59,-SPIRAL  GEARING, 


OF  THE 


116 


BROWN    &    SHARPE    MFG.    CO. 


If  we  now  turn  this  cylinder  A  B  C  D  in  the  direc- 
tion of  the  arrow,  the  spiral  Avill  advance  from  O  to  E. 
This  advance  is  the  pitch  of  the  spiral. 

The  angle  E  O  F,  which  the  spiral  makes  with  the 
axis  E  O,  is  the  angle  of  the  spiral.  This  angle  is 
found  as  in  Chapter  I.  The  circumference  of  the 
cylinder  corresponds  to  the  side  opposite  the  angle. 
The  pitch  of  the  spiral  corresponds  to  the  side  adjacent 
the  angle.  Hence  the  rule  for  getting  angle  of  spiral : 

Divide  the  circumference  of  the  cylinder  or  spiral 

i' ula ting    the 

parts  of  a  spi-  by  the  number  of  inches  of  spiral  to  o,i€  turn,  and  the 

rtil. 

quotient  will  be  the  tangent  of  angle  of  spiral. 

When  the  angle  of  spiral  and  circumference  are 
given,  to  find  the  pitch : 

Divide  the  circumference  by  the  tangent  of  angle, 
and  the  quotient  will  he  the  pitch  of  the  spiral. 

When  the  angle  of  spiral  and  the  lead  or  pitch  of 
spiral  are  given,  to  find  the  circumference  : 

Multiply  the  tangent  of  angle  by  the  pitch,  and  the 
product  will  be  the  circumference. 

When  applying  calculations  to  -spiral  gears  the  angle  is 
reckoned  at  the  pitch  circumference  and  not  at  the  outer  or 
addendum  circle. 

It  will  be  seen  that  when  two  spirals  of  different  diame- 
ters have  the  same  pitch  the  spiral  o^  less  diameter  will  have 
the  smaller  angle.  Thus  in  Fig.  60  if  the  paper  triangle  had 
been  4"  long  instead  of  G"  the  diameter  of  the  cylinder  would 
have  been  1.27"  and  the  angle  of  the  spiral  would  have  been 
only  ?>'2\  degrees. 


1'UOVIDENCE,    R.    I.  11 


CHAPTER  V. 

EXAMPLES  IN  CALCULATION  OF  PITCH   OF    SPIRAL-ANGLE  OF 

SPIRAL— CIRCUMFERENCE  OF   SPIRAL  GEARS- 

A  FEW  HINTS  ON  CUTTING. 


It  will  be  seen  that  the  rules  for  calculating  circum- 
ference of  spiral  gears,  angle  and  pitch  of  spiral  are 
the  same  as  in  Chapter  I,  for  tangent  and  angle  of  a  right 
angle  triangle.  In  Chapter  IV  the  word  .':  circumference" 
is  substituted  for '•  side  opposite,"  and  the  words  "pitch  of 
spiral"  are  substituted  for  side  •'  adjacent." 

When  two  spiral  gears  are  in  mesh  the  angle  of 
spiral  should  be  the  same  in  one  gear  as  in  the  other,  e?ce  to  Angle 

'  ot  Shafts. 

in  order  to  have  the  shafts  parallel  and  the  teeth  work 
properly  together.  When  two  gears  both  have  right 
hand  spiral  teeth,  or  both  have  left  hand  spiral  teeth, 
the  angle  of  their  shafts  will  be  equal  to  the  sum  of 
the  angles  of  their  spirals.  But  when  two  gears  have 
different  hand  spirals  the  angle  of  their  shafts  will  be 
equal  to  the  difference  of  their  angles  of  spirals. 
Thus,  in  Fig.  59  the  gears  A  and  B  both  have  right 
hand  spirals.  The  angle  of  both  spirals  is  45°,  their 
sum  is  90°,  or  their  axes  are  at  right  angles.  But  C 
has  a  left  hand  spiral  of  45°.  Hence,  as  the  difference 
between  angles  of  spirals  of  B  and  C  is  0,  their  axes 
are  parallel. 

When  the  two  gears  have  the  same  number  of  teeth 
the  pitch  of  the  spiral  will  be  alike  in  both  gears.    But 
when  one  gear  has  more  teeth  than  the  other  the  pitch 
of  spiral  in  the  larger  gear  ^should  be  longer  in  the 
same  ratio.     Thus,  if  one  gear  has  50  teeth  and  the    pitch  m  Spx 
other  gear  has  25  teeth,  the  pitch  of  spiral  in  the  50  Diameters.1  el 
tooth  wheel  should  be  twice  as  long  as  that  of  the  25 


BROWN    &    SHARPE   MFG.    CO. 

tooth  wheel.  Of  course,  the  diameter  of  pitch  circle 
should  be  twice  as  large  in  the  50  tooth  as  in  the  25 
tooth  wheel. 

In  spirals  where  the  angle  is  45°  the  circumference 
is  the  same  as  the  spiral  pitch,  because  the  tangent  of 
45°  is  1. 

Sometimes  the  circumference  is  varied  to  suit  a 
to  suit  A  spiral,  pitch  that  can  be  cut  011  the  machine  and  retain  the 
angle  required.  This  would  apply  to  cutting  rolls  for 
making  diamond-shaped  impressions  where  the  diam- 
eter of  the  roll  is  not  a  matter  of  importance. 

When  two  gears  are  to  run  together  in  a  given 
velocity  ratio,  it  is  well  to  first  select  spirals  that  the 
machine  will  cut  of  the  same  ratio,  and  calculate  the 
numbers  of  teeth  and  angle  to  correspond.  This  will 
often  save  considerable  time  in  figuring. 

The  calculations  for  spiral  gears  present  no  special 
difficulties,  but  sometimes  a  little  ingenuity  is  requiied 
to  make  work  conform  to  the  machine  and  to  such 
cutters  as  we  may  have  in  stock.  It  is  a  good  plan  to 
make  a  trial  piece  for  each  gear,  and  to  cut  a  few  teeth 
in  each  trial  piece  to  test  the  setting  of  the  machine. 
Dummiss  or  These  trial  pieces  are  called  "dummies."  If  the  gears 

Trial  Pisces.  to 

are  likely  to  be  duplicated,  each  dummy  can  be  marked 
and  kept  for  future  setting  of  the  machine.  Stamp  all 
the  data  on  the  dummies  ;  it  is  better  to  spend  a  little 
time  in  marking  dummies  than  a  good  deal  of  time 
hunting  up,  or  trying  to  remember,  old  data. 

Let  it  be  required  to  make  two  spiral  gears  to  run 
with  a  ratio  of  4  to  1,  the  distance  between  centers  to 
be  3.125"  (3J"). 

By  rule  given  in  Chapter  XII.,  Part  I.,  \ve  find  the 
diameters  of  pitch  circles  will  be  5"  and  1J".  Let  us 
take  a  spiral  of  48"  pitch  for  the  large  gear,  and  a 
spiral  of  12"  pitch  for  the  small  gear.  The  circumfer- 
ence of  the  5"  pitch  circle  is  15.70796".  Dividing 
the  circumference  by  the  pitch  of  the  spiral,  we  have 
15^L^L?_6— . 32724"  for  tangent  of  angle  of  spiral.  In 
the  table  the  nearest  angle  to  tangent,  .32724",  is  18°  10'. 
As  before  stated,  the  angle  of  the  teeth  in  the  small 
gear  will  be  the  same  as  the  angle  of  teeth  or  spiral  in 


PROVIDENCE,    K.    L  110 

the  large  gear.     Now,  this  rule  gives  the  angle  at  the .  A  difference 

pitch  surface  only.     Upon  looking  at  a  small  screw  of  £nd  Bottom  of 
*  Spiral  (Grooves, 

coarse  pitch,  it  will  be  seen  that  the  angle  at  bottom 

of  the  thread  is  not  so  great  as  the  angle  at  top  of 
thread ;  that  is,  the  thread  at  bottom  is  nearer  parallel 
to  the  center  line  than  that  at  the  top. 


This  will  be  seen  in  Fig.  61,  where  A  O  is  the  center 
line  ;  b  f  shows  direction  of  bottom  of  the  thread,  and 
d  g  shows  direction  of  top  of  thread.  The  angle  A/'  b 
is  less  than  the  angle  A  g  d,  This  difference  of  angle 
is  due  to  the  warped  nature  of  a  screw  thread,  and 
sometimes  makes  it  necessary  to  change  the  angle  for 
setting  work  from  the  figured  angle,  wrhen  a  rotary 
disk  cutter  is  used,  to  prevent  the  cutter  from  marring 
the  groove  as  the  teeth  of  cutter  enter  and  leave. 
How  much  to  change  the  angle  can  be  seen  by  inspec- 
tion when  cutting  the  dummies.  The  change  of  angle 
will  be  more  in  a  small  gear  of  a  given  pitch  than  in  a 
large  gear  of  the  same  pitch. 

A  rotary  disk  cutter  is  generally  preferable,  because  Bisk-cutters, 
it  cuts  faster  and  holds  its  shape  better.  Yet  it  is 
hardly  practical  to  cut  low  numbered  pinions  with 
rotary  disk  cutters,  because  for  some  distance  below 
the  pitch  line  the  spaces  are  so  nearly  parallel.  A  part 
of  the  difficulty  can  be  removed  by  making  the  cutter 
as  small  as  is  consistent  with  strength.  Still  more  of 
the  trouble  can  be  done  away  with  by  making  a  cutter 
on  a  shank,  the  center  of  the  work  and  the  center  of  ^  Shank  or  End 

Cutter. 

shank  cutter  then  being  in  the  same  plane.  When 
using  a  shank  cutter  the  center  of  the  work  is  perpen- 
dicular to  the  center  of  the  cutter,  no  adjustment  for 


BROWN    &    SHARPE    MFG.    CO. 

angle  being  made.  Strictly,  a  shank  mill  does  not  re- 
produce its  own  shape  in  cutting  a  spiral  groove.  In 
using  a  shank  cutter,  more  care  is  necessary  to  see 
that  the  work  does  not  slip.  It  may  be  well  to  rough 
out  with  a  disk  cutter  and  finish  with  a  shank  cutter. 
There  is  not  generally  much  difficulty  in  involute  or 
single-curve  spiral  gears  with  disk  cutters, 
o*  A  cylinder  2"  diameter  is  to  have  spiral  grooves  20° 
Pitch  of  spiral.  with  the  center  line  of  cylinder;  what  will  be  the  pitch 
of  spiral?  The  circumference  is  6.2832".  The  tan- 
gent of  20°  is  .36397.  Dividing  the  circumference  by 
the  tangent  of  angle,  we  obtain  6^||-||T=17.26"  +  for 
pitch  of  spiral. 

Before  cutting  into  a  blank  it  is  well  to  make  a 
slight  trace  of  the  spiral,  with  the  cutter,  after  the 
machine  is  geared  up,  to  see  if  the  gears  are  properly 
arranged.  Attention  to  this  may  avoid  spoiling  ;i 
blank. 

The  cutting  of  spiral  gears  develops  some  curious 
facts  to  one  who  may  not  have  studied  warped  sur- 
faces. 

In  the  Universal  Milling  Machine  wre  can  cut  a  class 
of  warped  surfaces  that  will  fit  a  straight  edge  in  two 
directions.  Thus,  in  Fig.  61,  if  it  we^e  possible  to  re- 
duce the  diameter  of  screw  and  then  cut  the  thread 
clear  down  to  the  center  line  A  O,  the  bottom  of  the 
thread  would  be  a  straight  line  running  through  the 
center  or  the  line  A  O  itself.  The  sides  would  still  be 
straight  as  in  the  figure.  If  we  should  cut  a  spiral 
groove  with  a  plain  rotary  disk  cutter,  having  parallel 
sides,  the  shape  of  the  grooves  would  have  but  little 
resemblance  to  that  of  the  cutter.  Taking  advantage 
of  this  principle,  we  learned  the  fact  that  spiral  gears 
can  be  planed  with  a  rack  tool. 
Spiral  Gears  The  gears,  Fig.  59,  were  planed.  The  tool  was  of 

cut  with  Rack 

Tool.  the  same  shape  as  the  spaces  in  the  rack  D  D.     All 

spiral  gears  of  the  same  pitch  could  be  planed  with 
one  tool. 

The  nature  of  this  can  be  seen  when  we  consider 
that  straight  rack  teeth  can  mesh  with  spiral  gears,  as 
in  Fig.  58. 


PROVIDENCE,    R.    I.  121 

We  have  succeeded  in  cutting  small  spiral  gears  with 
a  long  fly  tool,  cutting  on  one  side  only.  The  shape 
of  this  fly  tool  was  like  a  common  lathe  side  tool.  In 
this  case,  of  course,  the  gears  had  to  be  reversed  in 
order  to  finish  both  sides  of  teeth.  A  description  and  an 
illustration  of  cutting  spiral  and  spur  gears  with  a  fly  tool  on 
our  Universal  Milling  Machine  are  in  the  Amcrn'nit,  Machin- 
ist for  Nov.  21,  1885. 


BROWN    fc   SHAKPE   MTO.    CO. 


CHAPTER     VI. 

NORMAL  PITCH  OF  SPIRAL  GEARS— CURVATURE  OF  PITCH 
SURFACE— FORM  OF  CUTTERS. 


Normal  to  a     A  Normal  to  a  curve  is  a  line  perpendicular  to  the 
tangent  at  the  point  of  tangency. 


Fig.  62 


In  Fig.  62,  the  line  B  C  is  tangent  to  the  arc  D  E  F, 
and  the  line  A  E  O,  being  perpendicular  to  the  tan- 
gent at  E,  the  point  of  tangency,  is  a  normal  to  the 
arc. 

Fig.  63  is  a  representation  of  the  pitch  surface  of  a 
spiral  gear.  A'  D'  C'  is  the  circular  pitch,  as  in  Part 
I.  A  D  C  is  the  same  circular  pitch  seen  upon  the 
periphery  of  a  wheel.  Let  A  D  be  a  tooth  and  D  C  a 
space.  Now,  to  make  this  space  D  C,  the  path  of  cut- 
ting is  along  the  dotted  line  a  b.  By  mere  inspection, 
we  can  see  that  the  shortest  distance  between  two 
teeth  along  the  pitch  surface  is  not  the  distance 
ADC. 

Let  the  line  A  E  B  be  perpendicular  to  the  sides  of 
teeth  upon  the  pitch  surface.  A  continuation  of  this 
line,  perpendicular  to  all  the  teeth,  is  called  the 
Normal  Helix.  The  line  A  E  B,  reaching  over  a 
tooth  and  a  space  along  the  normal  helix,  is  called  the 
Normal  Pitch. 


PROVIDENCE,  It.  I. 


123 


Fifj. 


124  BROWN    &    SHARPE    MFG.    CO. 

Normal  Pitch.  The  Normal  Pitch  of  a  spiral  gear  is  then:  77ie 
shortest  distance  betioeen  the  centers  of  two  consecutive 
teeth  measured  along  the  pitch  surface. 

In  spur  gears  the  normal  pitch  and  circular  pitch 
are  alike.  In  the  rack  D  D,  Fig.  58,  the  linear  pitch 
and  normal  pitch  are  alike. 

s  Ijrau?ear8°r  From  tne  foregoing  it  will  be  seen  that,  if  we  should 
cut  the  space  D  C  with  a  cutter,  the  thickness  of  which 
at  the  pitch  line  is  equal  to  one-half  the  circular  pitch, 
as  in  spur  wheels,  the  space  would  be  too  wide,  and 
the  teeth  would  be  too  thin.  Hence,  spiral  gears 
should  be  cut  with  thinner  cutters  than  spur  gears  of 
the  same  circular  pitch. 

The  angle  C  A  B  is  equal  to  the  angle  of  the  spiral. 
The  line  A  E  B  corresponds  to  the  cosine  of  the  angle 
CAB.  Hence  the  rule :  Multiply  the  cosine  of  angle 
ma?  Pitch  Nor~°y  spiral  by  the  circular  pitch,  and  the  product  will  be 
the  normal  pitch.  One-half  the  normal  pitch  is  the 
proper  thickness  of  cutter  at  the  pitch  line. 

If  the  normal  pitch  and  the  angle  are  known  Divide  the. 
normal  pitch  by  the  cosine  of  tJ/e  angle  and  the  quotient 
will  be  the  linear  pitch. 

This  may  be  required  in  a  case  of  a  spiral  pinion  run- 
ning in  a  rack.  The  perpendicular  to  the  fcide  of  the  rack 
is  taken  as  the  line  from  which  to  calculate  angle  of  teeth. 
That  is,  this  line  would  correspond  to  the  axial  line  in  spiral 
gears.  This  considers  a  rack  as  a  gear  of  infinitely  long- 
radius  ;  page  12.  If  the  condition  required  gives  the  angle 
of  axis  of  gear  and  the  side  of  the  rack,  we  subtract  the 
given  angle  from  1)0  degrees  and  base  our  caculations  upon 
the  remainder,  which  is  cnwplciui'nt  of  tho  given  angle. 

The  addendum  and  working  depth  of  tooth  should 
correspond  to  the  normal  pitch,  and  not  to  the  circular 
pitch.  Thus,  if  the  normal  pitch  is  12  diametral,  the 
addendum  should  be  Ty,  the  thickness  .1309",  and  so 
on.  The  diameter  of  pitch  circle  of  a  spiral  gear  is 
calculated  from  the  diametral  pitch.  Thus  a  gear  of 
30  teeth  10  P  would  be  3"  pitch  diameter. 

But  if  the  normal  pitch  is  12  diametral  pitch,  the 
blank  will  be  '>TV   diameter  instead  of  3-=^". 
varies™*1  ^^     ^  *s  ev^en^  that  *ue  normal  pitch  varies  with  the 


PROVIDENCE,    K.    I.  125 

angle  of  spiral.  The  cutter  should  be  for  the  normal 
pitch.  In  designing  spiral  gears,  it  is  well  to  first 
look  over  list  of  cutters  on  hand,  and  see  if  there  are 
cutters  to  which  the  gears  can  be  made  to  conform. 
This  may  avoid  the  necessity  of  getting  a  new  cutter, 
or  of  changing  both  drawing  and  gears  after  they  are 
under  way  To  do  this,  the  problem  is  worked  the 
reverse  of  the  foregoing ;  that  is  : 

First  calculate  to  the  next  liner  pitch  cutter  than    To    m  a  u  « 

Angle  ot  Spir.i! 

would  be  required  for  the  diametral  pitch.  conform  to cut- 

•*•  A  tei'8  given. 

Let  us  take,  for  example,  a  gear  10  pitch  arid  30 
teeth  spiral.  Let  the  next  finer  cutter  be  for  12  pitch 
gears.  The  first  thing  is  to  find  the  angle  that  will 
make  the  normal  pitch  .2618",  when  the  circular  pitch 
is  .3142".  See  table  of  tooth  parts.  This  means  (Fig. 
63)  that  the  line  ADC  will  be  .3142"  when  A  E  B  is 
.2618".  Dividing  .2618"  by  .3142"  (see  Chapter  1V.\ 
we  obtain  the  cosine  of  the  angle  CAB,  which  is  also 
the  angle  of  the  spiral,  rff^f =-833. 

The  same  quotient  comes  by  dividing  10  by  12. 
•}-$=.  833  +  .  Looking  in  the  table,  we  find  the  angle 
corresponding  to  the  cosine  .833  is  33°  30'.  "We  now 
want  to  find  the  pitch  of  spiral  that  will  give  angle  of 
33 £°  on  t-he  pitch  surface  of  the  wheel,  3"  diameter. 
Dividing  the  circumference  by  the  tangent  of  angle, 
we  obtain  the  pitch  of  spiral  (see  Chapter  V.)  The 
circumference  is  9.4248".  The  tangent  of  33°  30'  is 
.66188,  ^J-ffl-g  =14.23;  and  we  have  for  our  spiral 
14.23"  lead. 

When  the  machine  is  not  arranged  for  the  exact    When   exact 

Pitch  cannot  ba 

pitch  of  spiral  wanted,  it  is  generally  well  enough  to  cut. 
take  the  next  nearest  spiral.  A  half  of  an  inch  more 
or  less  in  a  spiral  10"  pitch  or  more  would  hardly  be 
noticed  in  angle  of  teeth.  It  is  generally  better  to 
take  the  next  longer  spiral  and  cut  enough  deeper  to 
bring  center  distances  right.  When  two  gears  of  the 
same  size  are  in  mesh  with  their  axes  parallel,  a  change 
of  angle  of  teeth  or  spiral  makes  110  difference  in  the 
correct  meshing  of  the  teeth. 

But  when  shears  of  different  size  are  in  mesh,  duo    .sP.V'al  Ciears 

to  m  '  of   Different 

regard  must  be  had  to  the  spirals  being  in  pitch,  pro- sizes  to  Mesh 


J2G  BROWN    fc   SHARPS    MFG.    CO. 

portional  to  their  angular  velocities  (see  Chapter  Y . ) 
We  come  now  to  the  curvature  of  cutters  for  spiral 
gears;  that  is,  their  shape  as  to  whether  a  cutter  is 
made  to  cut  12  teeth  or  100  teeth.  A  cutter  that  is  right, 
shape  of  Cut-  to  cut  a  spur  gear  3"  diameter,  may  not  be  right  for  a 
spiral  gear  3"  diameter.  To  find  the  curvature  of 
cutter,  fit  a  templet  to  the  blank  along  the  line  of  the 
normal  helix,  as  A  E  B,  letting  the  templet  reach  over 
about  two  or  three  normal  pitches.  The  curvature  of 
this  templet  will  be  nearer  a  straight  line  than  an  arc 
of  the  addendum  circle.  Now  find  the  diameter  of  a 
circle  that  will  fit  this  templet,  and  consider  this  circle 
as  the  addendum  circle  of  a  gear  for  which  we  are  to 
select  a  cutter,  reckoning  the  gear  as  of  a  pitch  the 
same  as  the  normal  pitch. 


Fig.  64. 


Thus,  in  Fig.  64,  suppose  the  templet  fits  a  circle 
3J"  diameter,  if  the  normal  pitch  is  12  to  inch,  dia- 
metral, the  cutter  required  is  for  12  P  and  40  teeth. 
The  curvature  of  the  templet  will  not  be  quite  circular, 
but  is  sufficiently  near  for  practical  purposes.  Strictly, 
a  flat  templet  cannot  be  made  to  coincide  with  the 
normal  helix  for  any  distance  whatever,  but  any  greater 
refinement  than  we  have  suggested  can  hardly  be 
earned  out  in  a  workshop. 


PROVIDENCE,    K.    I.  ]  L'7 

This  applies  more  to  an  end  cutter,  for  a  disk  cutter  may 
have  the  right  shape  for  a  tooth  space  and  still  round  off 
the  teeth  too  much  on  account  of  the  warped  nature  of 
the  teeth. 

The  difference  between  normal  pitch  and  linear  or 
circular  pitch  is  plainly  seen  in.  Figs  58  and  59. 

The  rack  D  D,  Fig.  58,  is  of  regular  form,  the  depth 
of  teeth  being  -}-J  of  the  circular  pitch,  nearly  (.6866  of 
the  pitch,  accurately).  If  a  section  of  a  tooth  in  either 
of  the  gears  be  made  square  across  the  tooth,  that  is  a 
normal  section ,  the  depth  of  the  tooth  will  have  the 
same  relation  to  the  thickness  of  the  tooth  as  in  the 
rack  just  named. 

But  the  teeth  of  spiral  gears,  looking  at  them  upon 
the  side  of  the  gears,  are  thicker  in  proportion  to  their 
depth,  as  in  Fig.  59.  This  difference  is  seen  between 
the  teeth  of  the  two  racks  D  D  and  E  E,  Fig.  58.  In 
the  rack  D  D  we  have  20  teeth,  while  in  the  rack  E  E 
we  have  but  14  teeth ;  yet  each  rack  will  run  with  each 
of  the  spiral  gears  A,  B  or  C,  Fig.  59,  but  at  different 
angles. 

The  teeth  of  one  rack  will  accurately  fit  the  teeth  of 
the  other  rack  face  to  face,  but  the  sides  of  one  rack 
will  then  be  at  an  angle  of  45°  with  the  sides  of  the 
other  rack.  At  F  is  a  guide  for  holding  a  rack  m  mesh 
with  a  gear. 

The  reason  the  racks  will  each  run  with  either  of  the  three 
gears  is  because  all  the  gears  and  racks  have  the  same  normal 
pitch.  When  the  spiral  gears  are  to  run  together  they  must 
both  have  the  same  normal  pitch.  Hence  two  spiral  gears 
may  run  correctly  together  though  the  circular  pitch  of  one 
gear  is  not  like  the  circular  pitch  of  the  other  gear. 


128  BROWN    &   SHARPE   MF9.    00. 


CHAPTER  VII. 
SCREW  GEARS  AND  SPIRAL  GEARS— GENERAL   REMARKS, 


s  riraHiears  °f  ^^e  wol'kmg  °^  spiral  gears  is  generally  smoother 
than  spur  gears.  A  tooth  does  not  strike  along  its 
whole  face  or  length  at  once.  Tooth  contact  first  takes 
place  at  one  side  of  the  gear,  passes  across  the  face 
and  ceases  at  the  other  side  of  the  gear.  This  action 
tends  to  cover  defects  in  shape  of  teeth  and  the  adjust- 
ment of  centers. 

Since  the  invention  of  machines  for  producing  accu- 
rate epicyloidal  and  involute  curves,  it  has  not  so  often 
been  found  necessary  to  resort  to  spiral  gears  for 
smoothness  of  action.  A  greater  range  can  be  had  in 
the  adjustment  of  centers  in  spiral  gears  than  in  spur 
gears.  The  angle  of  the  teeth  should  be  enough,  BO 
that  one  pair  of  teeth  will  not  part  contact  at  one  side 
of  the  gears  until  the  next  pair  of  teeth  have  met  on  the 
other  side  of  the  gears.  When  this  is  done  the  gears 
will  be  in  mesh  so  long  as  the  circumferences  of  their 
addendum  circles  intersect  each  other.  This  is  some- 
times necessary  in  roll  gears. 

Eelative  to  spur  and  bevel  gears  in  Part  I.,  Chapter 
XII.,  ib  was  stated  that  all  gears  finally  wore  them- 
selves out  of  shape  and  might  become  noisy.  Spiral 
gears  may  be  worn  out  of  shape,  but  the  smoothness 
of  action  can  hardly  be  impaired  so  long  as  there  are 
any  teeth  left.  For  every  quantity  of  wear,  of  course, 
there  will  be  an  equal  quantity  of  backlash,  so  that  if 
gears  have  to  be  reversed  the  lost  motion  in  spiral 
gears  will  be  as  much  as  in  any  gears,  and  may  be 
u ilon1  shafts1  of more  ^  there  is  end  play  of  the  shafts.  In  spiral  gears 
spiral  (Jears.  there  is  end  pressure  upon  the  shafts,  because  of  the 
screw-like  action  of  the  teeth.  This  end  pressure  is 
sometimes  balanced  by  putting  two  gears  upon  each 
shaft,  one  of  right  and  one  of  left  hand  spiral. 


PROVIDENCE,    E.    I.  ^ 

The  same  result  is  obtained  in  solid  cast  gears  by 
making  the  pattern  in  two  parts — one  right  and  one 
left-hand  spiral.  Such  gears  are  colloquially  called 
"herring-bone  gears." 

In  an  internal  spiral  gear  and  its  pinion,  the  spirals 
of  both  wheels  are  either  right-handed  or  left-handed. 
Such  a  combination  would  hardly  be  a  mercantile 
product,  although  interesting  as  mechanical  feat. 

In  screw  or  worm-gears  the  axes  are  generally  at 
right  angles,  or  nearly  so.  The  distinctive  features  of 
screw  gearing  may  be  stated  as  follows : 

The  relative  angular  velocities  do  not  depend  upon 
the  diameters  of  pitch- cylinders,  as  in  Chapter  I., 
Part  I.  Thus  the  worm  in  Chapter  XL,  Fig.  35,  can.  Distinctive* 

features    of 

be   any  diameter — one   inch   or  ten   inches — without  Screw  Gearing- 

affecting  the  velocity  of  the  worm-wheel.      Conversely  if  the 

axes  are  not  parallel  we  can  have  a  pair  of  spiral  or  screw 

gears   of  the  same  diameter,  but  of  different  numbers  of 

teeth.     The  direction  in  which  a  worm-wheel  turns  depends 

upon  whether  the  worm  has  a  right-hand  or  left-hand  thread. 

When  angles  of    axes  of  worm  and  worm-wheel  are 

oblique,  there  is  a  practical   limit   to  the  directional 

relation   of    the   worm-wheel.      The  rotation  of   the 

worm-wheel   is   made  by  the    end   movement  of   the 

worm-thread. 

The  term  worm  and  worm-wheel,  or  worm-gearing, 
is  applied  to  cases  where  the  worms  are  cut  in  a  lathe. 

If  we  let  two  cylinders  touch  each  other,  their  axes 
be  at  right  angles,  the  rotation  of  one  cylinder  will 
have  no  tendency  to  turn  the  other  cylinder,  as  in 
Chapter  I.,  Part  I. 

We  can  now  see  why  worms  and  worm-wheels  wear  WjJJjJ 
out  faster  than  other  gearing.     The  length  of  worm-  so  fast, 
thread,  equal  to  more  than  the  entire  circumference  of 
worm,  comes  in  sliding  contact  with  each  tooth  of  the 
wheel  during  one  turn  of  the  wheel. 

The  angle  of  a  worm-thread  can  be  calculated  the 
same  as  the  angle  of  teeth  of  spiral  gear. 


B,10W.<    &.    SHARPE    MFG.    CO. 


CHAPTER  VIII. 

CONTINUED  FRACTIONS— SOME  APPLICATIONS   IN    MACHINE 
v     CONSTRUCTION. 


a  DconUnnue°d      ^  continued  fraction  is  one  which  has  unity  for  its 
Fraction.          numerator,  and  for  its  denominator  an  entire  number 
plus  a  fraction,  which  fraction  has  also  unity  for  its 
numerator,  and  for  its  denominator  an  entire  number 
plus  a  fraction,  and  thus  in  order. 
The  expression,  I 

3+T 

5  is  called  a  continued  frac- 
tion.    By  the  use  of  continued  fractions,  we  are  eiia- 
Practicai  use  bled  to  find  a  fraction  expressed  in  smaller  numbers, 
Fractions.         that,  for  practical  purposes,  may  be  sufficiently  near  in 
value  to  another  fraction  expressed  in  large  numbers. 
If  we  were  required  to  cut  a  worm  that  would  mesh 
with  a  gear  4  diametral  pitch  (4  P.),  in  a  lathe  having 

3  to  1-inch  linear  leading  screw,  we  might,  without 
continued  fractions,  have   trouble  in  finding  change 
gears,  because    the    circular   pitch  corresponding  to 

4  diametral  pitch  is    expressed  in   large   numbers : 

4  P=TVo5A  V. 

This  example  will  be  considered  farther  on.  For 
illustration,  we  will  take  a  simpler  example. 

What  fraction  expressed  in  smaller  numbers  is  near- 
est in  value  to  -££$1  Dividing  the  numerator  and  the 
denominator  of  a  fraction  by  the  same  number  does 
not  change  the  value  of  the  fraction.  Dividing  both 

co^u^ue1?*61*1118  of   T2*V  by  29'  we  liave    fcc~   or>    what  is  the 

same  thing  expressed  as  a  continued  fraction,  s-t-  1  .  The 

i 

continued  fraction  5+JL  is  exactly  equal  to  j2^.      If 

now,  we  reject  the  -fa,  the  fraction  -J  will  be  larger 
than  «-rJJr»  because  the  denominator  has  been  dimin- 
ished,  5  being  less  than  o^V      i  is    something  near 
T2j^    expressed    in    smaller    numbers    than    29    for   a 


PROVIDENCE,    R.    I.  131 

numerator  and  146  for  a  denjminator.  Reducing  -J 
and  -ffa  to  a  common  denominator,  we  have  ^-r=-^J$- 
and  iVe-^TTo-  Subtracting  one  from  the  other,  we 
have  T£¥,  which  is  the  difference  between  -J-  and  -ffo. 
Thus,  in  thinking  of  -ffa  as  -J-,  we  have  a  pretty  fair 
idea  of  its  value. 

There  are  fourteen  fractions  with  terms  smaller  than 
29  and  146,  which  are  nearer  -ffo  than  -J-  is,  sr.cli  as 
•iU|,  -|f-  and  so  on  to  f2fT.  In  this  case  by  continued  frac- 
tions we  obtain  only  one  approximation,  namely  -J,  and 
any  other  approximations,  as  -JJ-J-,  -Jf-,  &c.,  we  find  by 
trial.  It  will  be  noted  that  all  these  approximations 
are  greater  in  value  than  -ffj.  There  are  cases,  how- 
ever, in  which  we  can,  by  continued  fractions,  obtain 
approximations  both  greater  and  less  than  the  required 
fraction,  and  these  will  be  the  nearest  possible  approxi- 
mations that  there  can  be  in  smaller  terms  than  the 
given  fraction. 

In  the  French  metric  system,  a  millimetre  is  equal 
to  .03937  inch;  what  fraction  in  smaller  terms  ex- 
presses .03937"  nearly?  .03937,  in  a  vulgar  fraction, 
is  T-| ftjj-Jrj-.  Dividing  both  numerator  and  denominator 
by  3937,  we  have  £^4!.  Rejecting  from  the  de- 
nominator of  the  new  fraction,  -J-g-J4>  the  fraction  ^ 
gives  us  a  pretty  good  idea  of  the  value  of  .03937". 
If  in  the  expression,  25+111.5,  we  divide  both  terms  of 
the  fraction  -Jf -J4  by  1575,  the  value  will  not  be  changed. 
Performing  the  division,  we  have 


i 

25  +  1 


2  +  787 
1575- 

We  can  now  divide  both  terms  of  f-ffy  by  787, 
without  changing  its  value,  and  then  substitute  the 
new  fraction  for  17T8T\  ^n  *De  continued  fraction. 

Dividing  again,  and  substituting,  we  have  : 


25+_l  _ 

~2  +  l^_ 
8+  1 
787 

as   the   continued   fraction   that   is   exactly  equal   to 
.03937. 


132  BROWN    &    SHAKPE    MFG.    CO. 


In  performing  the  divisions,  the  work  stands  thus  i 

3937)  100000  (25 
7874 
21260 
196&5 

1575)  3937  (2 
3150 

787)  1575  (2 
1574 

1)  787  (787 
787 

•o- 

That  is,  dividing  the  last  divisor  by  the  last  remain- 
der, as  in  finding  the  greatest  common  divisor.  The 
quotients  become  the  denominators  of  the  continued 
fraction,  with  unity  for  numerators.  The  denominators 
25,  2,  and  so  on,  are  called  incomplete  quotients,  since 
they  are  only  the  entire  parts  of  each  quotient.  The 
first  expression  in  the  continued  fraction  is  ^  or 
.04  —  a  little  larger  than  .03937.  If,  now,  we  take 
we  shall  come  still  nearer  .03937.  The  expres- 


sion j^qij:  is  merely  stating  that  1  is  to  be  divided  by 
25£.  To  divide,  we  first  reduce  25£  to  an  improper 
fraction,  ^-,  and  the  expression  becomes  IT,  or  one 
divided  by  ^-.  To  divide  by  a  fraction,  "Invert  the 
divisor,  and  proceed  as  in  multiplication."  We 
then  have  -^2T  as  the  next  nearest  fraction  to  .03937. 
-^=.0392  +  ,  which  is  smaller  than  .03937.  To  get  still 
nearer,  we  take  in  the  next  part  of  the  continued  frac- 
tion, and  have 


25  + 


2  +  1 
2' 

We  can  bring  the  value  of  this  expression  into  a 
fraction,  with  only  one  number  for  its  numerator  and 
one  number  for  its  denominator,  by  performing  the 
operations  indicated,  step  by  step,  commencing  at  the 
last  part  of  the  cootinued  fraction.  Thus,  2-f  -J-,  or 
2£,  is  equal  to  -f ,  Stopping  here,  the  continued  frac- 
tion would  become  _l_  * 

25+J_ 
5 

T- 

1  \ 

Now,  jf~  equals  f,  and  we  have  25  +  2.      25|  equals- 

2  5 

J-f-1 ;  substituting  again,  we  have  iii.     Dividing  1  by 
!$!,  we  have  T|T.      T|T  is   the   nearest   fraction  to 


PBOVIDENCE,    R.    I.  133 

.03937,  unless  we  reduce  the  whole  continued  fraction 
i 

25  +  1 

2  +  1 


2  +  L-j  which  would  give  us  back  the  .03937  itself. 

787 

TfT=. 03937007,  which  is  only  Timrjimnr  larger 
.03937.  It  is  not  often  that  an  approximation  will 
•come  so  near  as  this. 

This  ratio,  5  to  127,  is  used  in  cutting  millimeter     Practical  use 

,  of  the  foregoing 

thread  screws.  If  the  leading  screw  of  the  lathe  is  Example. 
1  to  one  inch,  the  change  gears  will  have  the  ratio  of 
5  to  127;  if  8  to  one  inch,  the  ratio  will  be  8  times 
as  large,  or  40  to  127;  so  that  with  leading  screw  8  to 
inch,  and  change  gears  40  and  127,  we  can  cut  milli- 
meter threads  near  enough  for  practical  purposes. 

The  foregoing  operations   are  more  tedious  in  de- 
scription than  in  use.     The  steps  have  been  carefully 
n  »tod,   so   that  the  reason  for  each  step  can  be  seen 
from  rules  of  common  arithmetic,  the  operations  being 
merely  reducing  complex  fractions.     The  reductions, 
^j,  -fT<  yHJhp  etc.,  are  called  conver gents,  because  they 
•come  nearer  and  nearer  to  the  required  .03937.     The 
operations  can  be  shortened  as  follows: 

Let  us  find  the  fractions  converging  towards  .7854",  Example, 
the  circular  pitch  of  4  diametral  pitch,  .7854=T'y\r5T54¥; 
reducing  to  lowest  terms,  we  have  -fffj.      Applying 
ihe  operation  for  the  greatest  common  divisor: 

3927)  5000  (1 
8927 

1073)  3927  (3 
3219 

708)  1073  (1 
_708 

365)  708  (1 
365 

843)  365  (1 
343 

22)  343  (15 
22 
123 

no 

13)  22  (1 
13 
-9)13  <1 

~4)  9  (2 
8 

1)  4  (4 
0 

Bringing  the  various  incomplete  quotients  as  de- 
nomiuators  in  a  continued  fraction  as  before,  we  hnve: 


134  BROWN    &    SHAEPE    MFG.    CO. 

1 


__ 

1  +  1 


15  +  1  _ 

1  +  1  _ 

1+1  _ 

•+1 

Now  arrange  each  partial  quotient  in  a  line,  thus  r 

13111       15        11         2  4 

1    I    I     *    tt    Ht    Iff    HI    TST¥T    MM 

Now  place  under  the  first  incomplete  quotient  the* 
first  reduction  or  convergent  ^,  which,  of  course,  is  1  ; 
put  under  the  next  partial  quotient  the  next  reduction  or 
convergent  \  —  r  or  ~,  which  becomes  f  . 

1  is  larger  than  .7854>  and  j  is  less  than  .7854. 

Having  made  two  reductions,  as  previously  shown, 
we  can  shorten  the  operations  by  the  following  rule  for  next 
convergents:  Multiply  the  numerator  of  the  convergent 
just  found  by  the  denominator  of  the  next  term  of  the  con- 
tinued fraction,  or  the  next  incomplete  quotient,  and  add 
to  the  product  the  numerator  of  the  preceding  convergent; 
the  sum  will  be  the  numerator  of  the  next  convergent. 

Proceed  in  the  same  way  for  the  denominator,  that 

is  multiply  the  denominator  of   the   convergent   just 

found  by  the  next  incomplete  quotient  and  add  to  the 

product  the  denominator  of  the  preceding  convergent  ; 

-the  sum  will  be  the  denominator  of  the  next  convergent. 

Continue  until  the  last  convergent  is  the  original  frac- 

tion.   Under  each  incomplete  quotient  or  denominator 

from  the  continued  fraction  arranged  in  line,  will  be 

seen  the  corresponding  convergent  or  reduction.     The 

convergent  |J  is  the  one  commonly  used  in  cutting 

racks  4  P.     This  is  the  same  as  calling  the  circumference  of 

a  circle  22-7  when  the  diameter  is  one  (1)  ;  this  is  also  the 

common  ratio  for  cutting  any  rack.     The  equivalent  decimal 

to  -J-J  is  .7857  X  ,  being  about  -3-5-$^  large.     In  three  set- 

tings for  rack  teeth,  this  error  would  amount  to  about  .001" 

For  a  worm,  this  corresponds  to  ^  threads  to  1"; 
now,  with  a  leading  screw  of  lathe  3  to  1",  we  would 
want  gears  on  the  spindle  and  screw  in  a  ratio  of  33 
to  14. 

Hence,  a  gear  on  the  spindle  with  66  teeth,  and  a 
gear  on  the  3  threa  1  screw  of  28  teeth,  would  enable 
us  to  cut  a  worm  to  fit  a  4  P  gear. 


PROVIDENCE,    E.    I. 


137 


CHAPTER   IX. 
ANGLE    OF    PRESSURE. 


In  Fig.  47,  let  A  be  any  flat  disk  lying  upon  a  hori- 
zontal plane.  Take  any  piece,  B,  with  a  square  end, 
a  b.  Press  against  A  with  the  piece  B  in  the  direction 
of  the  arrow. 


Fig.  65. 


Fig.  66. 


It  is  evident  A  will  tend  to  move  directly  ahead  of 
B  in  the  normal  line  c  d.  Now  (Fig.  66)  let  the  piece 
B,  at  one  corner/1,  touch  the  piece  A.  Move  the  piece 
B  along  the  line  d  e,  in  the  direction  of  the  arrow. 

It  is  evident  that  A  will  not  now  tend  to  move  in 
the  line  d  e,  but  will  tend  to  move  in  the  direction  of 
the  normal  c  d.  When  one  piece,  not  attached,  presses 
against  another,  the  tendency  to  move  the  second 
piece  is  in  the  direction  of  the  normal,  at  the  point  of 
contact.  This  normal  is  called  the  line  of  pressure. 
The  angle  that  this  line  makes  with  the  path  of  the 
impelling  piece,  is  called  the  angle  of  pressure. 

In  Part  I.,  Chapter  IV.,  the  lines  B  A  and  B  A'  are 
called  lines  of  pressure  This  means  that  if  the  gear 


Une  of  Press- 


138 


BROWN    &    SHARPE    MFG.    CO. 


drives  the  rack,  the  tendency  to  move  the  rack  is  not 
in  the  direction  of  pitch  line  of  rack,  but  either  in  the 
direction  B  A  or  B  A',  as  we  turn  the  wheel  to  the  left 
or  to  the  right. 

The  same  law  holds  if  the  rack  is  moved  in  the 
direction  of  the  pitch  line ;  the  tendency  to  move  the 
wheel  is  not  directly  tangent  to  the  pitch  circle,  as  if 
driven  by  a  belt,  but  in  the  direction  of  the  line  of 
pressure.  Of  course  the  rack  and  wheel  do  move  in 
the  paths  prescribed  by  their  connections  with  the 
framework,  the  wheel  turning  about  its  axis  and  the 
rack  moving  along  its  ways.  This  pressure,  not  in  a 
direct  path  of  the  moving  piece,  causes  extra  friction 
in  all  toothed  gearing  that  cannot  well  be  avoided. 

Although  this  pressure  works  out  by  the  diagram, 
as  we  have  shown,  yet,  in  the  actual  gears,  it  is  not  at 
all  certain  that  they  will  follow  the  law  as  stated, 
because  of  the  friction  of  teeth  among  themselves.  If 
the  driver  in  a  train  of  gears  has  no  bearing  upon  its 
tooth-flank,  we  apprehend  there  will  be  but  little 
tendency  to  press  the  shafts  apart. 
Arc  of  Action.  The  arc  through  which  a  wheel  passes  while  one  of 

its  teeth  is  in  contact  is  called  the  arc  of  action. 
tema8of°ime^     Until  witnin  a  few  years,  the  base  of  a  system  of 
Ge<?rs.geable double-curve  interchangeable  gears  was  12  teeth.     It 
is  now  15  teeth  in  the  best  practice  (see  Chapter  VII., 
Part  I.) 

The  reason  for  this  change  was  :  the  base,  15  teeth, 
gives  less  angle  of  pressure  and  longer  arc  of  contact, 
and  hence  longer  lifetime  of  gears. 


PKOVTDENCE,    B.    T.  139 


CHAPTER  X. 
INTERNAL    GEARS, 


In  Part  I.,  Chapter  VIII.,  it  was  stated  that  the 
space  of  an  internal  gear  is  the  same  as  the  tooth  of  a 
spur  gear.  This  applies  to  involute  or  single-curve 
gears  as  well  as  to  double-curve  gears. 

The  sides  of  teeth  in  involute  internal  gears  will  be 
hollowing.  It,  however,  has  been  customary  to  cut 
internal  gears  with  spur  gear-cutters,  a  No.  1  cutter 
generally  being  used.  This  makes  the  teeth  sides 
convex.  Special  cutters  should  be  made  for  coarse  Special  Cut. 

..,,,,  T      ^      .        .         .     .  .  ters  for  coarse 

pitch  double-curve  gears.  In  designing  internal  gears,  Pitch, 
it  is  sometimes  necessary  to  depart  from  the  system 
with  15-tooth  base,  so  as  to  have  the  pinion  differ  from 
the  wheel  by  less  than  15  teeth.  The  rules  given  in 
Part  I.,  Chapters  VII.  and  VIII.,  will  apply  in  making 
gears  on  any  base  besides  15  teeth.  If  the  base  is 
low-numbered  and  the  pinion  is  small,  it  may  be  neces- 
sary to  resort  to  the  method  given  at  the  end  of  Chap- 
ter VII.,  because  the  teeth  may  be  too  much  rounded 
at  the  points  by  following  the  approximate  rules. 

The  base  must  be  as  small  as  the  difference  between    Base  for  in- 
ternal    Gear 
the  internal  gear  and   its  pinion.     The  base  can  be  Teeth. 

smaller  if  desired. 

Let  it  be  required  to  make  an  internal  gear,  and 
pinion  24  and  18  teeth,  3  P.  Here  the  base  cannot 
be  more  than  6  teeth. 

In  Fig.  67  the  base  is  6  teeth.  The  arcs  A  K  and 
O  k,  drawn  about  T,  have  a  radius  equal  to  the  radius 
of  the  pitch  circle  of  a  6-tooth  gear,  3  P,  instead  of  a 
15-tooth  gear,  as  in  Chapter  VIII.,  Part  I. 

The  outline  of  teeth  of   both  gears  and  pinion  is    Description  of 
made  similar  to  the  gear  in  Chapter  VIII.      The  same 


140 


BROWN    &    SHARPE    MFG.    CO. 


GEAR,  24  TEETH. 
PINION,  18  TEETH,  3  P. 


.  67. 

JA 


N  =24  and  1  8 
P'=  1.0472* 
t=-    5236* 
8=.     .3333* 
D"=     .6666" 
8+/=     .3857* 
'+/=.     .7190' 


INTERNAL  GEAR  AND   PINION    IN    MESH, 


PROVIDENCE,    11.    I. 

letters  refer  to  similar  parts.  The  clearance  circle  is, 
however,  drawn  on  the  outside  for  the  internal  gear. 
As  before  stated,  the  spaces  of  a  spur  wheel  become 
the  teeth  of  an  internal  wheel.  The  teeth  of  internal 
gears  require  but  little  for  fillets  at  the  roots;  they 
are  generally  strong  enough  without  fillets.  The 
teeth  of  the  pinion  are  also  similar  to  the  gear  in 
Chapter  VIII.,  substituting  6-tooth  for  15-tooth  base. 
To  avoid  confusion,  it  is  well  to  make  a  complete 
sketch  of  one  gear  before  making  the  other.  The  arc 
of  action  is  longer  in  internal  gears  than  in  external 
gears.  This  property  sometimes  makes  it  necessary 
to  give  less  fillets  than  in  external  gears. 

In  Fig.  67  the  angle  K  T  A  is  30°  instead  of  12°,  as 
in  Fig.  12.  This  brings  the  line  of  pressure  L  P  at 
an  angle  of  60°  with  the  radius  C  T,  instead  of  78°. 
A  system  of  spur  gears  could  be  made  upon  this 
6-tooth  base.  These  gears  would  interchange,  but  no 
gear  of  this  6-tooth  system  would  mesh  with  a  double- 
curve  gear  made  upon  the  15-tooth  system  in  Part  1. 


142 


BROWN    &    SHARPE    MFG.    CO. 


CHAPTER  XI. 


STRENGTH  OF  GEARING. 


We  have  been  unable  to  derive  from  our  own  experi- 
ence, any  definite  rule  on  this  subject,  but  would  refer 
those  intere-ted  to  u  Kent's  Mechanical  Engineers' 
Pocket  Book,"  where  a  good  treatment  of  the  subject 
can  be  found. 

We  give  a  few  examples  of  average  breaking  strain 
of  our  Combination  Gears,  as  determined  by  dyna- 
mometer, the  pressure  being  measured  at  the  pitch  line. 
These  gears  are  of  cast  iron,  with  cut  teeth. 


DIAMETRAL    PITCH. 

REVOLUTIONS 

PRESSURE  AT 

No.  TEETH. 

PER  MINUTE. 

PITCH  LINE. 

•        FACE. 

10 

1   1-16 

110 

27 

1060 

8 

1   1-4 

72 

40 

1460 

6 

1   9-16 

72 

27 

2220 

5 

1  7-8 

90 

18 

2470 

These  arc  the  actual  pressures  for  the  particular 
widths  given. 

If  we  take  a  safe  pressure  at  1-3  of  the  foregoing 
breaking  strain,  we  shall  have  for 


10  Pitch  853  1-3  Lbs.  at  the  Pitch  Line. 
8     "      486  2-3  "  " 

6     "      740  "  " 

5     "      823  1-3  "  " 

The  width  of  the  face  of  a  gear  is  in  good  proportion 
when  it  is  2-J  times  the  circular  pitch.  Brown  & 
Sharpe's  rule  is,  width—  J- -f. 25" 


PROVIDENCE,  R.  I. 


143 


TABLE  OF  DECIMAL  EQUIVALENTS 

OF  MILLIMETERS  AND  FRACTIONS  OF  MILLIMETERS. 


mm.  Inches. 

mm.   Inches. 

mm.   Inches. 

-gV=.  00079 

||-=.  02047 

2=  .07874 

•ft  =.00157 

f|=  .  02126 

3=  .11811 

V\r=.  00236 

|f  =  .  02205 

4=  .15748 

Tfr=.  00315 

|J=  .  02283 

5=  .19685 

1fir=.  00394 

J=  .  02362 

6=  .23622 

-ft  =  .00472 

i=.  02441 

7=  .27559 

3V=.  00551 

J=  .  02520 

8=  .31496 

-^  =.00630 

M=.  02598 

9=  .35433 

A=-  00709 

U=.  02677 

10=  .39370 

if  =.00787 

|f  =.02756 

11=  .43307 

|i=.  00866 

f£=.  02835 

12=  .47244 

|f  =.00945 

U=.  02913 

13=  .51181 

|f=.  01024 

|f  =  .  02992 

14=  .55118 

-H-=.  01102 

|f  =.03071 

15=  .59055 

t*=-  01181 

if  =  03150 

16=  .62992 

|f=  01260 

ff=.  03228 

17=  .66929 

-if:=  01339 

|f=  03307 

18=  .70866 

J|=.  01417 

-|f  =  .  03386 

19=  .74803 

if=  01496 

tt=  03465 

20=  .78740 

I"  =  .01575 

4f=.  03543 

21=  .82677 

-|J  =  .  01654 

|f  =.03622 

22=  .86614 

ff=.  01732 

|f  =.03701 

23=  .90551 

if=.  01811 

ff=.  03780 

24=  .94488 

|*=.  01890 

|f  =.03858 

25=  .98425 

||=  .  01969 

1=.  03937 

26=1.02362 

10  mm. -I  Centimeter=0.3937  inches. 
10  cm.  =1  Decimeter  =3.937        " 
10  dm.  =1  Meter          =39.37        " 
25.4  mm.=l  English  Inch. 


144 


BROWN    &    SHARPS   MFG.    CO. 


NATUKAL  SINE. 


Des. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

.00000 

.00291 

.00581 

.00872 

.01103 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02326 

.02617 

.02908 

.03199 

.03489 

88 

2 

.03489 

.03780 

.04071 

.04361 

.04652 

.04943 

.05233 

87 

3 

.05233 

.05524 

.05814 

.06104 

.06395 

.06685 

.06975 

86 

4 

.06975 

.07265 

.07555 

.07845 

.081S5 

.08425 

.08715 

85 

5 

.08715 

.09005 

.09295 

.09584 

.09874 

.10163 

.10452 

84 

G 

.10452 

.10742 

.11031 

.11320 

.11609 

.11898 

.12186 

83 

7 

.12186 

.12475 

.12764 

.13052 

.13341 

.  13629 

.13917 

82 

8 

.13917 

.14205 

.14493 

.14780 

.15068 

.15356 

.15643 

81 

9 

.15643 

.15930 

.16217 

.16504 

.16791 

.17078 

.17364 

80 

10 

.  17364 

.17651 

.17937 

.18223 

.18509 

.18795 

.19080 

79 

11 

.19080 

.19366 

.19651 

.19936 

.20221 

.20506 

.20791 

78 

12 

.20791 

.21075 

.21359 

.21644 

.21927 

.22211 

.22495 

77 

13 

.22495 

.22778 

.23061 

.23344 

.23627 

.23909 

.24192 

76 

14 

.24192 

.24474 

.24756 

.25038 

.25319 

.25600 

.25881 

75 

15 

.25881 

.261G2 

.26443 

.26723 

.27004 

.27284 

.27563 

74 

10 

.27563 

.27843 

.28122 

.28401 

.28680 

.28958 

.29237 

73 

17 

.29237 

.29515 

.29793 

.30070 

.30347 

.30624 

.30901 

72 

18 

.30901 

.31170 

.31454 

.31730 

.32006 

.32281 

.32556 

71 

19 

.32556 

.32831 

.33106 

.33380 

.33654 

.33928 

.34202 

70 

20 

.34202 

.34475 

.34748 

.35020 

.35293 

.35565 

.35836 

69 

21 

.35830 

.36108 

.36379 

.36650 

.36920 

.37190 

.37460 

60 

22 

.37460 

.37780 

.37999 

.38268 

.38536 

.38805 

.39073 

67 

23 

.39073 

.39340 

.39607 

.39874 

.40141 

.40407 

.40673 

66 

24 

.40673 

.409£9 

.41204 

.41469 

.41733 

.41998 

.42261 

65 

25 

.42261 

.42525 

.42788 

.43051 

.43313 

.43575 

.43837 

54 

26 

.43837 

.44098 

.44359 

.44619 

.44879 

.45139 

.45399 

63 

27 

.45399 

.45658 

.45916 

.46174 

.46432 

.46690 

.46947 

G2 

28 

.46947 

.47203 

.47460 

.47715 

.47971 

.48226 

.48481 

61 

29 

.48481 

.48735 

.48989 

.49242 

.49495 

.49747 

.50000 

60 

30 

.50000 

.50251 

.50503 

.50753 

.51004 

.51254 

.51503 

59 

31 

.51503 

.51752 

.52001 

.52249 

.52497 

.52745 

.52991 

58 

32 

.52991 

.53238 

.53484 

.53730 

.53975 

.54219 

.54463 

57 

33 

.54463 

.54707 

.54950 

.55193 

.55436 

.55677 

.55919 

56 

34 

.55919 

.56160 

.56400 

.56640 

.56880 

.57119 

.57357 

55 

35 

.57357 

.57595 

.57833 

.58070 

.58306 

.58542 

.58778 

54 

36 

.58778 

.59013 

.59248 

.59482 

.59715 

.59948 

.60181 

!  53 

37 

.60181 

.60413 

.60645 

.60876 

.61106 

.61336 

.61566 

52 

38 

.61566 

.61795 

.62023 

.62251 

.62478 

.62705 

.62932 

51 

39 

.62932 

.63157 

.63383 

.63607 

.63832 

.64055 

.64278 

50 

40 

.64278 

.64501 

.64723 

.64944 

.65165 

.65386 

.65605 

49 

41 

.65605 

.65825 

.66043 

.66262 

.66479 

.66696 

.G6913 

48 

42 

.66913 

.67128 

.67344 

.67559 

.67773 

.67986 

.68199 

47 

43 

.68199 

.68412 

.68624 

.68835 

.69046 

.69256 

.69465 

46 

44 

.69465 

.69674 

.69883 

.7009J 

.70298 

.70504 

.70710 

45 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Dez. 

NATUKAL   COSINE. 


PROVIDENCE,   R.    I. 


NATURAL  SINE. 


Deg. 

0'       10' 

20' 

30' 

40' 

50' 

60' 

45 

.70710 

.70916 

.71120 

.71325 

.71528 

.71731 

.71934 

44 

48 

.71934 

.72135 

.72336 

.72537 

.72737 

.72936 

.73135 

43 

47 

.73135 

.73333 

.  73530 

.73727 

.73923 

.74119 

.74314 

42 

48 

.74314 

.74508 

.74702 

.74895 

.75088 

.75279 

.75471 

41 

49 

.75471 

.75661 

.75851 

.  76040 

.76229 

.76417 

.76604 

40 

50 

.76604 

.76791 

.76977 

.771G2 

.77347 

.77531 

.77714 

89 

51 

.77714 

.77897 

.78079 

.78260 

.78441 

.78621 

.78801 

38 

53 

.78801 

.78979 

.79157 

.79335 

.  79512 

.79688 

.  79863 

37 

53 

.79863 

.80038 

.80212 

.80385 

.80558 

.80730 

.80901 

36 

54 

.80901 

.81072 

.81242 

.81411 

.81580 

.81748 

.81915 

35 

55 

.81915 

.82081 

.82247 

.82412 

.82577 

.82740 

.82903 

34 

56 

.82903 

.83066 

.83227 

.83388 

.83548 

.83708 

.83867 

33 

57 

.83867 

.84025 

.'84182 

.84339 

.84495 

.84650 

.84804 

32 

53 

.84804 

.84958 

.85111 

.85264 

.85415 

.85566 

.85716 

31 

59 

.85716 

.85866 

.86014 

.86162 

.88310 

.86456 

.86602 

30 

CO 

.86602 

.86747 

.86892 

.87035 

.87178 

.87320 

.87462 

29 

Gl 

.87462 

.87602 

.87742 

.87881 

.88020 

.88157 

.88294 

28 

63 

.88294 

.88430 

.88566 

.88701 

.88835 

.88968 

.89100 

27 

03 

.89100 

.89232 

.89363 

.89493 

.89622 

.89751 

.89879 

26 

64 

.89879 

.90006 

.90132 

.90258 

.90383 

.90507 

.90630 

25 

65 

.90630 

.90753 

.90875 

.90996 

.91116 

.91235 

.91354 

24 

66 

-91354 

.91472 

.91589 

.91706 

.91821 

.91936 

.92050 

23 

67 

.92050 

.92163 

.92276 

.92388 

.92498 

.92609 

.92718 

22 

68 

.92718 

.92827 

.92934 

.93041 

.93148 

.93253 

.93358 

21 

69 

.93358 

.93461 

.93565 

.93667 

.93768 

.93869 

.93969 

20 

70 

.93969 

.94068 

.94166 

.94264 

.94360 

.94456 

.94551 

19 

71 

.94551 

.94646 

.94789 

.94832 

.94924 

.95015 

.95105 

18 

72 

.95105 

.95195 

.95283 

.95371 

.95458 

.95545 

.95630 

17 

73 

.95630 

.95715 

.95799 

.95882 

.95964 

.96045 

.96126 

16 

74 

.96126 

.96205 

.96284 

.96363 

.96440 

.96516 

.96592 

15 

75 

.96592 

.96667 

.96741 

.96814 

.96887 

.96958 

.97029 

14 

76 

.97029 

.97099 

.97168 

.97237 

.97304 

.97371 

.97437 

13 

77 

.97437 

.97502 

.97566 

.97620 

.97692 

.97753 

.97814 

12 

78 

.97814 

.97874 

.97934 

.97992 

.98050 

.98106 

.98162 

11 

79 

.98162 

.98217 

.98272 

.98325 

.98378 

.98429 

.98480 

10 

80 

.98480 

.98530 

.98580 

.98628 

.98676 

.98722 

.98768 

9 

81 

.98768 

.98813 

.98858 

.98901 

.98944 

.98985 

.99026 

8 

82 

.99026 

.99066 

.09106 

.99144 

.99182 

.99218 

.99254 

7 

83 

.99254 

.99289 

.99323 

.99357 

.99389 

.99421 

.99452 

6 

84 

.99452 

.99482 

.99511 

.99539 

.99567 

.99593 

.99619 

5 

85 

.99619 

.99644 

.99668 

.99691 

.99714 

.99735 

.99756 

4 

86 

.99756 

.99776 

.99795 

99813 

.99830 

.99847 

.99863 

3 

87 

.99863 

.99877 

.99891 

.99904 

.99917 

.99928 

.99939 

2 

88 

.99939 

.99948 

.99957 

.99965 

.99972 

.99979 

.99984 

1 

89 

.99984 

.99989 

.99993 

.99996 

.99998 

.99999 

1.0000 

0 

- 

60' 

50' 

40' 

30'      20' 

10' 

0' 

Dcg. 

NATURAL   COSINE. 


146 


BROWX  &  SHARPE  MFG.  CO, 


NATURAL  TANGENT. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

.00000 

.00290 

.00581 

.00872 

.01163 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02327 

.02618 

.02909 

.03200 

.03492 

88 

2 

.03492 

.03783 

.04074 

.04366 

.04657 

.04949 

.05240 

87 

3 

.05240 

.05532 

.05824 

.06116 

.06408 

.06700 

.06992 

86 

4 

.06992 

.07285 

.07577 

.07870 

.08162 

.08455 

.08748 

85 

5 

.08748 

.09042 

.09335 

.09628 

.09922 

.10216 

.10510 

84 

6 

.10510 

.10804 

.11099 

.11393 

.11688 

.11983 

.12278 

83 

7 

.12278 

.12573 

.12869 

.13165 

.13461 

.13757 

.14054 

83 

8 

.14054 

.14350 

.14647 

.14945 

.15242 

.15540 

.15838 

81 

9 

.15838 

.16136 

.16435 

.16734 

.17033 

.17332 

.17632 

80 

10 

.17632 

.17932 

.18233 

.18533 

.18834 

.19136 

.19438 

79 

11 

.19438 

.  19740 

.20042 

.20345 

.20648 

.20951 

.21255 

78 

12 

.21255 

.21559 

.21864 

.22169 

.22474 

.22780 

.23086 

77 

13 

.23086 

.23393 

.23700 

.24007 

.24315 

.24624 

.24932 

76 

14 

.24932 

.25242 

.25551 

.25861 

.26173 

.26483 

.26794 

75 

15 

.26794 

.27106 

.27419 

.27732 

.28046 

.28360 

.28674 

74 

16 

.28674 

.28989 

.29305 

.29621 

.29938 

.30255 

.30573 

73 

17  1 

.30573 

.30891 

.31210 

.31529 

.31850 

.32170 

.32493 

72 

13 

.32492 

.32813 

.33136 

.33459 

.33783 

.34107 

.34432 

71 

19 

.34482 

.34758 

.35084 

.35411 

.35739 

.36067 

.36397 

70 

20 

.36397 

.36726 

.37057 

.37388 

.37720 

.38053 

.38386 

69 

21 

.38386 

.38720 

.39055 

.39391 

.39727 

.40064 

.40402 

68 

23 

.40402 

.  40741 

.41080 

.41421 

.41762 

.43104 

.42447 

67 

23 

.42447 

.42791 

.43135 

.43481 

.43827 

.44174 

.44522 

60 

24 

.44522 

.44871 

.45221 

.45572 

.45924 

.46277 

.46630 

65 

25 

.46630 

.46985 

.47341 

.47697 

.48055 

.48413 

.48773 

64 

26 

.48773 

.49133 

.49495 

.49858 

.50221 

.50586 

.50952 

63 

27 

.50953 

.51319 

.51687 

.52056 

.52427 

.52798 

.53170 

62 

28  | 

.53170 

.53544 

.53919 

.54295 

.54672 

.55051 

.55430 

61 

29  ' 

.55430 

.55811 

.56193 

.56577 

.56961 

.57847 

.57735 

60 

30 

.57735 

.58123 

.58513 

.58904 

.59297 

.59690 

.60086 

59 

31 

.60086 

.60482 

.60880 

.61280 

.61680 

.63083 

.63486 

58 

32 

.62486 

.62892 

.63298 

.63707 

.64116 

.64528 

.64940 

57 

33 

.64940 

.65355 

.65771 

.66188 

.66607 

.67028 

.67450 

56 

34 

.67450 

.67874 

.68300 

.68728 

.69157 

.69588 

.70030 

55 

35 

.70020 

.70455 

.70891 

.71329 

.71769 

.72210 

.73654 

54 

36 

.72654 

.73099 

.73546 

.73996 

.74447 

.74900 

.75355 

53 

37 

.75355 

.75812 

.76271 

.76732 

.77195 

.77661 

.78138 

52 

38 

.78128 

.78598 

.79069 

.79543 

.80019 

.80497 

80978 

51 

39 

.80978 

.81461 

.81946 

.82433 

.82923 

.83415 

.83910 

50 

40 

.83910 

.84406 

.84906 

.85408 

.85912 

.86419 

.86928 

49 

41 

.86928 

.87440 

.87955 

.88473 

.88992 

.89515 

.90040 

48 

42 

.90040 

.90568 

.91099 

.91633 

.92169 

.92709 

.93251 

47 

43 

.93251 

.93796 

.94345 

.94896 

.95450 

.96008 

.96568 

40 

44 

.96568 

.97132 

.97699 

.98269 

.98843 

.99419 

1.0000 

45 

W 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT, 


PROVIDENCE,  Jl.  I, 


147 


NATURAL  TANGENT. 


Deg. 

o' 

10' 

20' 

80' 

40' 

50' 

60 

45 

1.0000 

1.0058 

1.0117 

1.0176 

1.0235 

1  0295 

1.0355 

44 

46 

1.0355 

1.0415 

1.0476 

1.0537 

1.0599 

1.0661 

1.0723 

43 

47 

1.0723 

1.0786 

1.0849 

1.0913 

1.0977 

1  .  1041 

1.1106 

42 

48 

1.1106 

1.1171 

1.1236 

1.1302 

1.1369 

1.1436 

1.1503 

41 

49 

1.1503 

1.1571 

1.1639 

1.1708 

1.1777 

1.1847 

1.1917 

40 

50 

1.1917 

1.1988 

1.2059 

1.2131 

1.2203 

1.2275 

1  2349 

39 

51 

1.2349 

1.2422 

1.2496 

1.2571 

1.2647 

1.2723 

1.2799 

38 

52 

1.2799 

1.2876 

1.2954 

1.3032 

1.3111 

1.3190 

1.3270 

37 

53 

1.3270 

.3351 

1.&4S2 

1.3514 

1.3596 

1.3680 

1.3763 

36 

54 

1.3763 

.3818 

1.3933 

1.4019 

1.4106 

1.4193 

1.4281 

35 

55 

1.4281 

.4370 

1.4459 

1.4550 

1.4641 

1.4733 

1.4825 

34 

56 

1  4825 

.4919 

1.5013 

1.5108 

1.5204 

1.5301 

1.5398 

33 

57 

1.5398 

.5497 

1.5596 

1.5696 

1.5798 

1.5900 

1.6003 

32 

58 

1.6003 

.6107 

1.6212 

1.6318 

1.6425 

1.6533 

1.6642 

31 

59 

1.6642 

6753 

1.6864 

1.6976 

1.7090 

1.7204 

1.7320 

30 

60 

1.7320 

1.7437 

1.7555 

1  .  7674 

1.7795 

1.7917 

1.8040 

29 

61 

1.8040 

1.8164 

1.8290 

1.8417 

1.8546 

1.8676 

1.8807 

28 

62 

1.8807 

1.8940 

1.9074 

1.9209 

1.9347 

1.9485 

1.9626 

27 

63 

1.9626 

1.9768 

1.9911 

2.0056 

2.0203 

2.0352 

2  0503 

26 

64 

2.0503 

2.0655 

2.0809 

2.0965 

2.1123 

2.1283 

2.1445 

25 

65 

2.1445 

2.1609 

2.1774 

2.1943 

2.2113 

2.2285 

2.2460 

24 

66 

2.2460 

2.2637 

2.2816 

2.2998 

2.3182 

2.3369 

2.3558 

23 

67 

2.3558 

2.3750 

2.3944 

2.4142 

2.4342 

2.4545 

2.4750 

22 

68 

2.4750 

2.4959 

2.5171 

2.5386 

2.5604 

2.5826 

2.6050 

21 

69 

2.6050 

2.6279 

2.6510 

2.6746 

2.6985 

2.7228 

2.7474 

20 

70 

2.7474 

2.7725 

2.7980 

2.8239 

2.8502 

2.8770 

2.9042 

19 

71 

2.9042 

2.9318 

2.9600 

2.9886 

3.0178 

3.0474 

3.0776 

18 

72 

3.0776 

3.1084 

3.1397 

3.1715 

3.2040 

3.2371 

3.2708 

17 

73 

3.2708 

3.3052 

3.3402 

3.3759 

3.4123 

3.4495 

3.4874 

16 

74 

3.4874 

3.5260 

3.5655 

3.6058 

3.6470 

3.6890 

3.7320 

15 

75 

3.7320 

3.7759 

3.8208 

3.8667 

3.9136 

3.9616 

4.0107 

14 

76 

4.0107 

4.0610 

4,1125 

4.1653 

4.2193 

4.2747 

4.3314 

13 

77 

4.3bl4 

4.3896 

4.4494 

4.5107 

4.5736 

4.6382 

4.7046 

12 

78 

4.7046 

4.7728 

4.8430 

4.9151 

4.9894 

5  065S 

5.1445 

11 

79 

5  .  1445 

5.2256 

5.3092 

5.3955 

5.4845 

5.5763 

5.6712 

10 

80 

5.6712 

5.7693 

5.8708 

5.9757 

6.0844 

6.1970 

6  3137 

9 

81 

6.3137 

6.4348 

6.5605 

6.6911 

6.8269 

6.9682 

7.1153 

8 

82 

7  1153 

7.2687 

7.4287 

7.5957 

7.7703 

7.9530 

8.1443 

7 

83 

8.1443 

8.3449 

8.5555 

8.7768 

9.0098 

9.2553 

9.5143 

6 

84 

9.5143 

9.7881 

10.078 

10.385 

10.711 

11.059 

11.430 

5 

85 

11.430 

11.826 

12.250 

12.706 

13.196 

13  726 

14.300 

4 

86 

14.300 

14.924 

15.604 

16  349 

17.169 

18.075 

19.081 

3 

87 

19.081 

20.205 

21.470 

22.904 

24.541 

26.431 

28.636 

2 

88 

28.636 

31.241 

34.367 

38.188 

42.964 

49.103 

57.290 

1 

89 

57.290 

68.750 

85.939 

114.58 

171.88 

343.77 

QO 

0 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


148 


BROWN   &    SHABPE   MFQ.    CO. 


NATUKAL   SECANT. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50 

60' 

. 

0 

.1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0001 

1.0001 

89 

1 

1.0001 

1  .  0002 

1.0002 

1.0003 

1.0004 

1.0005 

.1.0006 

88 

2 

1.0006 

1.0007 

1  .  0008 

1.0009 

1.0010 

1.0012 

1.0013 

87 

3 

1.0013 

1.0015 

1.0016 

1.0018 

1.0020 

1.0022 

1.0024 

86 

4 

1.0024 

1.0026 

1.0028 

1.0030 

1.0033 

1.0035 

1.0038 

85  | 

5 

1.0038 

1.0040 

1.0043 

1.0046 

1.0049 

1.0052 

1.0055 

84 

6 

1.0055 

1.0058 

1.0061 

1.0064 

1.0068 

1.0071 

1.0075 

83 

7 

1.0075 

1.0078 

1.0082 

1.0086 

1.0090 

1.0094 

1.0098 

82 

8 

1.0098 

1.0102 

1.0106 

1.0111 

1.0115 

1.0120 

1.0124 

81 

9 

1.0124 

1.0129 

1.0134 

1.0139 

1.0144 

1.0149 

1.0154 

80 

10 

1.0154 

1.0159 

1.0164 

1.0170 

1.0175 

1.0181 

1.0187 

79 

11 

1.0187 

1.0192 

1.0198 

1.0204 

1.0210 

1.0217 

1.0223 

78 

12 

1.0223 

1.0229 

1.0236 

1.0242 

1.0249 

1.0256 

1.0263 

77 

13 

1.0263 

1.0269 

1.0277 

1.0284 

1.0291 

1.0298 

1.0306 

76 

14 

1.0303 

1.0313 

.0321 

1.0329 

1.0336 

1.0344 

1.0352 

75 

15 

1.0352 

1.0360 

.0369 

1.0377 

1.0385 

1.0394 

1.0402 

74 

16 

1.0402 

1.0411 

.0420 

1.0429 

1.0438 

1.0447 

1.0456 

73 

17 

1.0456 

1.0466 

.0475 

1.0485 

1.0494 

1.0504 

1.0514 

72 

18 

.0514 

1.0524 

.0534 

1.0544 

1.0555 

1.0565 

1.0576 

71 

19 

!  .0576 

1.0586 

1.0597 

1.0008 

1.0619 

1.0630 

1.0641 

70 

20 

.0641 

1.0653 

1.0664 

1.0U76 

1.0087 

1.0699 

1.0711 

69 

21 

.0711 

1.0723 

1.0735 

1.0747 

1.0760 

1.0772 

1.0785 

68 

22 

.0785 

.0798 

1.0810 

1.0823 

1.0837 

1.0850 

1.0883 

67 

23 

1.0863 

.0877 

1.0890 

1.0904 

1.0918 

1.0932 

1.0946 

66  , 

24 

1.0946 

.0960 

1.0974 

1.0989 

1.1004 

1.1018 

1.1033 

65 

25 

1.1033 

.1048 

1.1063 

1.1079 

1.1094 

1.1110 

1.1126 

64 

26 

1.1126 

.1141 

1.1157 

1.1174 

1.1190 

1.1206 

1.1223 

63 

27 

1  .  1223 

1.1239 

1.1256 

1.1273 

1.1290 

1.1308 

1.1325 

62 

28 

1.1325 

1.1343 

1.1361 

1.1378 

1.1396 

1.1415 

1.1433 

61 

29 

1.1433 

1.1452 

1.1470 

1.1489 

1.1508 

1  .  1527 

1.1547 

60 

30 

1.1547 

1.1566 

1.1586 

1.1605 

1.1625 

1.1646 

1.1666 

59 

31 

1.1666 

1.1686 

1.1707 

1.1728 

1.1749 

1.1770 

1.1791 

58 

32 

1.1791 

1.1813 

1.1835 

1.1856 

1.1878 

1.1901 

1.1923 

57 

33 

1.1923 

1.1946 

1.1969 

1.1992 

1.2015 

1.20S8 

1.2062 

56 

34 

1.  20,52 

1.2085 

1.2109 

1.2134 

1.2158 

1.2182 

1.2207 

55 

35 

1.2207 

1.2232 

1.2257 

1.2283 

1.2308 

1.2334 

1.2360 

54 

31 

1.2360 

1.2386 

1.2413 

1.2440 

1.2466 

1.2494 

1.2521 

53 

37 

1.2521 

1.254S 

1.2576 

1.2504 

1.2632 

1.2661 

1.26UO 

52 

33 

1.2690 

1.2719 

1.2748 

1.2777 

1.2807 

1.2837 

1.2867 

51 

39 

1.2867 

1.2898 

1.2928 

1.2959 

1.2990 

1.3022 

1.3054 

50 

40 

1.3054 

1.3086 

1.3118 

1.3150 

1.3183 

1.3216 

1.3250 

49 

41 

1.32-;0 

1.3283 

1.3317 

1.3351 

1.3386 

1.3421 

1.3456 

48 

42 

1.3456 

1.3491 

1.3527 

1.3563 

1.3599 

1.3636 

1.3673 

47 

43  ! 

1  3673 

1.3710 

1.3748 

1.3785 

1.3824 

1.3862 

1.3901 

46 

44 

1.3901 

1.3940 

1.3980 

1.4020 

1.40JO 

1.4101 

1.4142 

45 

GO' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COSECANT. 


PROVIDENCE,    R.    I. 


149 


NATUKAL   SECANT. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

45 

1.4142 

1.4183 

1.4225 

1.4257 

1.4309 

1.4352 

1.4395 

^4 

46 

1.4395 

1.443J 

1.4483 

1.4527 

1.45721 

\  1.4817 

1.46t2 

43 

47 

1.4662 

1.4708 

1.4755 

1.4801 

1.48491 

'  1.4896 

1.4944 

42 

48 

1.4944 

1.4993 

1.5042 

1.5091 

1.5141,1 

1.5191 

1.5242 

41 

49 

1.5242 

1.52D3 

1.5345 

1.5397 

1.54501 

1.5503 

1.5557 

40 

50 

1.5557 

1.5611 

1.5666 

1.5721 

1.577f 

1.5833 

1.5890 

39 

51 

1.5890 

1.5947 

1.6005 

1.0063 

1.61V 

1.6182 

1.6242 

38 

52 

1.6242 

1.6303 

1.6364 

1.6426 

1.6jl9 

1.6552 

1.6616 

37 

53 

1.6616 

1.6680 

1.6745 

1.6811 

1.W78 

1.6945 

1.7013 

36 

54 

1.7013 

1.7081 

1.7150 

1.7220 

1.J291 

1.7362 

1.7434 

35 

55 

1.7434 

1.7507 

1.7580 

1.7655 

1/730 

1.7806 

1.78S2 

34 

56 

1.7882 

1.7960 

1.8038 

1.8118 

JT8198 

1.8278 

1.8860 

33 

57 

1.8360 

1.8443 

1.8527 

1.8611 

^8697 

1.8783 

1.8870 

32 

58 

1.8870 

1.8:059 

1.9048 

1.913SI 

^9230 

1.9322 

1.9416 

31 

59 

1.9416 

1.9510 

1.9608 

1.9703| 

F.9800 

1.9899 

2.0000 

30 

60 

2  0000 

2.0101 

2.0203 

2  0307 

2.0412 

2.0519 

2.0o26 

29 

01 

2.0626 

2.0735 

2.0845 

2.0957 

2.1070 

2.1184 

2.1300 

28 

62 

2.1300 

2.1417 

2.1536 

2.1656 

2.1778 

2.1901 

2.2026 

27 

63 

2.2026 

2.2153 

2.2281 

2.2411 

2.2543 

2.2(576 

2.2811 

26 

64 

2.2811 

2.2948 

2.3087 

2.3228 

2.3370 

2.3515 

2.3662 

25 

65 

2.3662 

2.3810 

2.3961 

2.4114 

2.4259 

2.4426 

2.4585 

24 

66 

2.4585 

2.4747 

2.4911 

2.5078 

2.5247 

2.5418 

2.5593 

23 

67 

2.5593 

2  5769 

2.5949 

2.6131 

2.6316 

2.6503 

2.6694 

22 

68 

2.6694 

2.6883 

2.7085 

2.7285 

2  7488 

2.7694 

2.7904 

21 

69 

2.7904 

2.8117 

2.8334 

2.8554 

2.8778 

2.9006 

2.9238 

20 

70 

2.9238 

2.9473 

2.9713 

2.9957 

3.0205 

3.0458 

3.0715 

19 

71 

3.0715 

3  0977 

3.1243 

3.1515 

3.1791 

3.2073 

3.2360 

18 

72 

3.2360 

3.2(553 

3.2951 

3.3255 

3.3564 

3.3880 

3.4203 

17 

73 

3.4203 

3.4531 

3.4867 

3.5209 

3  5558 

3.5915 

3.6279 

;  16 

74 

3.6279 

3.6651 

3.7031 

3.7419 

3.7816 

3  8222 

3.8637 

15 

75 

3.8037 

3.9061 

3  9495 

3.9939 

4.0393 

4.0859 

4.1335 

14 

76 

4.1335 

4.1823 

4.2323 

4  2836 

4.3362 

4.3901 

4.4454 

13 

77 

4.4454 

4.5021 

4.5604 

4.6202 

4.6816 

4.7448 

4.8097 

12 

78 

4.8097 

4.8764 

4.9451 

5  0158 

5.0886 

5.1635 

5.2408 

11 

79 

5.2408 

5.3204 

5.4026 

5.4874 

5.5749 

5.6653 

5.7587 

10 

80 

5.7587 

5.8553 

5.9553 

6.0588 

6.1660 

6.2771 

6.3924 

9 

81 

6.3924 

6.5120 

6.6363 

6.7654 

6.8997 

7.0396 

7.1852 

8 

82 

7.1852 

7.3371 

7.4957 

7.6612 

7.8344 

8.0156 

8.2055 

7 

83 

8.2055 

8.4046 

8.6137 

8.8336 

9.0651 

9.3091 

9.5667 

6 

84 

9.5667 

9.8391 

10.127 

10.433 

10.758 

11.104 

11.473 

5 

85 

11.473 

11.868 

12.291 

12.745 

13.234 

13.763 

14.335 

4 

86 

14.335 

14.957 

15.636 

16.380 

17.198 

18.102 

19.107 

3 

87 

19.107 

20.230 

21.493 

22.925 

24.562 

26.450 

28  653 

2 

88 

28.653 

31.257 

34.382 

38.201 

42  975 

49.114 

57.298 

1 

83 

57.298 

68.757 

85.945 

114.59 

171.88 

343.77 

00 

0 

GO' 

50" 

40' 

30' 

20' 

10' 

0' 

Deg. 

1 

NATURAL  COSECANT. 


150 


BROWN    &    SHARPE    MFG.    CO. 


TABLE    OF   DECIMAL   EQUIVALENTS 

OF  STHS,  16THS,  32NDS  AND  64THS  OF  AN  INCH. 


Sths. 

A=.  28125 

£f  =.296875 

J=.125 

#=.34375 

4f=.  328125 

{=.250 

§.40625 

f  f  =  .  359375 

$=.375 

.46875 

|f  =  .  390625 

.53125 

|J=.  421875 

£=.*625 

-lf=.  59375 

||=.  453125 

|=.750 

|J=.  65625 

|f  =  .  484375 

§=.876 

-||=.  71875 

-ff=.  515625 

16ths. 

^=.0625 
fV=.1875 

i|==.  78125 

-1|=  .  84375 
||=.  90625 
ft=.  96875 

|f  =.546875 
|-2  =  .578125 
|f=  .  609375 
ff=.  640625 

T<V=.3125 

TV=.4375 

64ths. 

f|=.  671875 
ff=.  703125 

1  6 

¥V=.  015625 

f  J=  734375 

!$  =  .  bo75 

A=-  046875 

£f=  765625 

-ff  =.8125 
if  =.9375 

A=.  078125 
-gV=.  109375 

j£=  .  796875 
ff=.  828125 

32nd$. 

A=-  140625 

|f  =  .  859375 

&  =.03125 

£±=.171875 

|.J=.  890625 

&=.  09375 

£f  =.203125 

ff=.  921875 

•&=.  15625 

£f=.  234375 

|f  =  .  953125 

fo=.  21875 

|}=.  265625 

|f  =.984375 

I  N  DEX. 


A. 

PAOE. 

Abbreviations  of  Parts  of  Teeth  and  Gears 4 

Addendum 2 

Angle,  How  to  Lay  Off  an 92, 1 11 

"      Increment 110 

"      of  Edge 106 

"      of  Face 108 

"      of  Pressure 137 

"      of  Spiral 117 

Angular  Velocity 2 

Annular  Gears 82,  1 39 

Arc  of  Action 138 

B. 

Base  Circle 11 

"     of  Epicycloidal  System 25 

"     of  Internal  Gears 139 

Bevel  Gear  Blanks. 34 

"           Cutting  on  B.  &  S.  Automatic  Gear  Cutter 52 

"           Angles  by  Diagram 36 

"  "      by  Calculation 106, 110 

"           Form  of  Teeth  of 41 

"  Whole  Diameter  of 36, 108 

C. 

Centers,  Line  of 2 

Circular  Pitch 4 

Classification  of  Gearing 5 

Clearance  at  Bottom  of  Space 6 

"         in  Pattern  Gears „ 8 

Condition  -of  Constant  Velocity  Katio 2 

Contact,  Arc  of 138 

Continued  Fractions 130 

Coppering  Solution 83 


1 54  .  INDEX. 

PAGE. 

Cutters,  How  to  Order 81 

"  Table  of  Epicycloidal 82 

"  of  Involute 80 

"  "  of  Speeds  for 79 

Cutting  Bevel  Gears  on  H.  &  S.  Automatic  Gear  Cuiter 52 

D. 

Decimal  Equivalents,  Tables  of 143, 1 50 

Diameter  Increment .' 108 

"         of  Pitch  Circle 6 

"         Pitch 5 

Diametral  Pitch 17 

Distance  between  Centers 8 

E. 

Elements  of  Gear  Teeth 5 

Epicycloidal  Gears,  with  more  and  less  than  15  Teeth 30 

"       with  15  Teeth 25 

Rack 27 

F. 

Face,  Width  of  Spur  Gear 78 

Flanks  of  Teeth  in  Low-numbered  Pinions 20 

G. 

Gear  Cutters,  How  to  Order 81 

"     Patterns 8 

Gearing  Classified 5 

Gears,  Bevel 34,41,106 

"      Epicycloidal 25 

"      Involute „ . .  9 

"      Spiral • 113 

"      Worm 62 

H. 

Herring-bone  Gears 129 

I. 

Increment,  Angle 110 

Diameter , 108 

Interchangeable  Gears 24 

Internal  or  Annular  Gears 139 

Involute  Gears,  30  Teeth  and  over 9 

"       with  Less  than  30  Teeth 20 

"       Rack.  .  12 


INDEX.  155 

L. 

PAGE. 

Limiting  Numbers  of  Teeth  in  Internal  Gears 32 

Line  of  Centers 

"     of  Pressure 12,  137 

Linear  Velocity 

M. 

Machine,  B.  &  S.,  for  Cutting  Bevel  Gears 52 

N. 

Normal 122 

"      Helix 122 

"      Pitch 122 

O. 

Original  Cylinders. 1 

P. 

Pattern  Gears 8 

Pitch  Circle 3 

"     Circular  or  Linear 4 

"     a  Diameter 6 

"     Diametral 17 

"     Normal 122 

"     of  Spirals 116 

Polygons,  Calculations  for  Diameters  of 99 

R. 

Back 12 

"    for  Epicycloidal  Gears 27 

"    for  Involute             "     12 

"    for  Spiral                 « 127 

Relative  Angular  Velocity 2 

Rolling  Contact  of  Pitch  Circle 3 

S. 

Screw  Gearing 113, 129 

Single-Curve  Teeth 9 

Speed  of  Gear  Cutters 79 


156  INDEX. 

PAGB. 

Spiral  Gearing 113 

Standard  Templets 27 

Strength  of  Gears .,.....« !42 

T. 

Table  of  Decimal  Equivalents 143, 150 

"      of  Sines,  etc 144 

u      of  Speeds  for  Gear  Cutters 79 

"      of  Tooth  Parts , 86 

V. 

Velocity,  Angular 2 

Linear 1 

"       Eelative. 2 

w. 

Wear  of  Teeth \ 78,128 

Worm  Gears..,  62 


~  OF  THE. 

'ITNIVERSITT 


.  OF  THE 

•UNIVERSITY 


LD  21-loOm-7,'33 


Scf*- 


t 


